pack all data files to data.json

App.py

@@ -1,6 +1,8 @@
 import random
 
 import gradio as gr
+import matplotlib.pyplot as plt
+
 from DPMInteractive import g_st, g_et, g_num, g_res
 from DPMInteractive import init_change, shrink_change, conv_change 
 from DPMInteractive import cond_prob_init_change, cond_prob_alpha_change, cond_prob_cond_change
@@ -353,44 +355,13 @@ def run_app():
             then(fore_param["method"], fore_param["inputs"], fore_param["outputs"], show_progress="minimal"). \
             then(ctr_param["method"], ctr_param["inputs"], ctr_param["outputs"], show_progress="minimal"). \
             then(fixed_param["method"], fixed_param["inputs"], fixed_param["outputs"], show_progress="minimal")
-
-    demo.launch(share=True, allowed_paths=["/"])
-        
-    return
-
-
-def gtx():
      
-    with gr.Blocks(css=g_css, head=js_head, js=js_load) as demo:
-        gr.Markdown("$$ $$", visible=False)
-        
-        md_introduction_block()
-
-        md_transform_block()
-
-        md_likelihood_block()
-
-        md_posterior_block()
-        
-        md_forward_process_block()
-        
-        md_backward_process_block()
-        
-        md_fit_posterior_block()
-        
-        md_posterior_transform_block()
-        
-        md_deconvolution_block()
-        
-        md_reference_block()
-        
-        md_about_block()
-    
-    demo.queue()
     demo.launch(allowed_paths=["/"])
+        
     return
-    
 
+    
 if __name__ == "__main__":
     run_app()
-    # gtx()
+    
+    

DPMInteractive.py

@@ -15,13 +15,12 @@ import matplotlib.pyplot as plt
 from scipy.spatial.distance import jensenshannon
 from scipy.optimize import curve_fit
 
-# import multiprocessing
-# from multiprocessing import Pool, Queue, Manager
+import multiprocessing
+from multiprocessing import Pool, Queue, Manager
 
 plt.rcParams['figure.constrained_layout.use'] = True
-plt.rcParams['figure.max_open_warning'] = 50
-plt.rcParams['interactive'] = False
-plt.ioff()
+plt.rcParams['figure.max_open_warning'] = 10
+matplotlib.rcParams['interactive'] = False
 
 
 g_st, g_et, g_num = -2.3, 2.3, 460
@@ -283,7 +282,8 @@ def init_change(seed, shrink_alpha, conv_alpha):
     init_fig, x, x_pdf = plot_init_pdf(seed, g_st, g_et, g_num)
     shrink_fig = plot_shrink_pdf(x, x_pdf, shrink_alpha, g_st, g_res)
     conv_fig = plot_conv_pdf(x, x_pdf, conv_alpha, g_res)
-    shrink_conv_fig = plot_shrink_conv_pdf(x, x_pdf, shrink_alpha, conv_alpha, g_st, g_res)
+    shrink_conv_fig = plot_shrink_conv_pdf(x, x_pdf, shrink_alpha, conv_alpha, g_st, g_res) 
+
     return init_fig, x, x_pdf, shrink_fig, conv_fig, shrink_conv_fig
 
 
@@ -515,6 +515,9 @@ def forward_seq_apply(x, x_pdf, st_alpha, et_alpha, step):
     
     # plot_forward_pdf(axes, seq_info, "blue")
     # plot_backward_bc_pdf(back_axes, seq_info)
+     
+    # fig.tight_layout()
+    # back_fig.tight_layout()
     
     forward_plot_state = fig, axes, pos_fig, pos_axes, seq_info, g_res, "blue"
     
@@ -771,6 +774,8 @@ def contraction_apply(x, x_pdf, bc_pdf, seed):
     axes[1].add_artist(axes[1].legend(handles[:2], labels[:2], handlelength=1.0, loc="upper left"))
     axes[1].add_artist(axes[1].legend(handles[2:], labels[2:], handlelength=0, loc="upper right"))
     
+    # fig.tight_layout()
+    
     return fig
 
 
@@ -790,6 +795,7 @@ def fixed_point_init_change(seed, x, x_pdf):
     axes[0].plot([], [], label="div=%0.2f"%div, color="orange")
     axes[0].legend(handlelength=1.2)
     
+    # fig.tight_layout()
     return fig, z, z_pdf, None
 
 

ExtraBlock.js

@@ -1,38 +1,49 @@
 
 
 
-function write_markdown() {
+async function write_markdown() {
     let names = ["introduction", "transform", "likelihood", "posterior", "forward_process", "backward_process",
                  "fit_posterior", "posterior_transform", "deconvolution", "reference", "about"];
-    names = names.slice(-1)
+    // names = names.slice(-1)
     
-    names.forEach((item, index) => {
-        const elem = document.getElementById("md_" + item);
-        let a = document.createElement('a');
-        a.href = "data:application/octet-stream," + encodeURIComponent(elem.outerHTML);
-        a.download = item + '.html';
-        a.click();
+    let data = await fetch("file/data.json").then(response => response.json());
+    
+    names.forEach((name, index) => {
+        let elem_zh = document.getElementById("md_" + name + "_zh");
+        if (elem_zh != null) { data[name+"_zh"] = elem_zh.outerHTML; }
+
+        const elem_en = document.getElementById("md_" + name + "_en");
+        if (elem_en != null) { data[name+"_en"] = elem_en.outerHTML; }
     });
+    
+    let a = document.createElement('a');
+    a.href = "data:application/octet-stream," + encodeURIComponent(JSON.stringify(data));
+    a.download = "data.json";
+    a.click();
 }
 
 
-function insert_markdown() {
-    const names = ["introduction", "transform", "likelihood", "posterior", "forward_process", "backward_process", 
-                   "fit_posterior", "posterior_transform", "deconvolution", "reference", "about"];
+async function insert_markdown() {
+    let names = ["introduction", "transform", "likelihood", "posterior", "forward_process", "backward_process", 
+                 "fit_posterior", "posterior_transform", "deconvolution", "reference", "about"];
+
+    let data = await fetch("file/data.json").then(response => response.json());
     
     for (let i = 0; i < names.length; i++) {
         name = names[i];
         const markdown_zh = document.createElement('div');
         markdown_zh.id = "md_" + name + "_zh";
         markdown_zh.style.display = "none";
-
-        fetch('file/Markdown/zh/' + name + ".html").then(response => response.text()).then(text => markdown_zh.innerHTML = text)
+        
+        markdown_zh.innerHTML = data[name+"_zh"] 
+        // fetch('file/Markdown/zh/' + name + ".html").then(response => response.text()).then(text => markdown_zh.innerHTML = text)
 
         const markdown_en = document.createElement('div');
         markdown_en.id = "md_" + name + "_en";
         markdown_en.style.display = "block";
-
-        fetch('file/Markdown/en/' + name + ".html").then(response => response.text()).then(text => markdown_en.innerHTML = text)
+        markdown_en.innerHTML = data[name+"_en"] 
+         
+        // fetch('file/Markdown/en/' + name + ".html").then(response => response.text()).then(text => markdown_en.innerHTML = text)
 
         const accord = document.getElementById(name).children[2];
         accord.appendChild(markdown_zh);
@@ -135,11 +146,59 @@ function katex_render(name) {
 }
 
 
-function insert_special_formula(name) {
+function insert_special_formula() {
     katex_render("zh_fit_0");
     katex_render("zh_fit_1");
     katex_render("zh_fit_2");
     katex_render("en_fit_0");
     katex_render("en_fit_1");
     katex_render("en_fit_2");
+}
+
+
+// async function insert_one_figure(name) {
+//     await fetch("file/" + name + ".png").then(response => response.blob()).then(blob => {
+//         const reader = new FileReader();
+//         reader.readAsDataURL(blob);
+//         return new Promise(() => {reader.onloadend = () => {
+//             const b64 = reader.result;
+//             let elem_en = document.getElementById("en_" + name);
+//             if (elem_en != null) {
+//                 elem_en.src = b64;
+//             }
+//             let elem_zh = document.getElementById("zh_" + name);
+//             if (elem_zh != null) {
+//                 elem_zh.src = b64;
+//             }
+//         }
+//         } )
+//     })
+// }
+
+async function insert_one_figure(name) {
+    await new Promise(r => setTimeout(r, 3000));
+    
+    let elem;
+    let elem_en = document.getElementById("en_" + name);
+    if (elem_en != null) { elem = elem_en; }
+    
+    let elem_zh = document.getElementById("zh_" + name);
+    if (elem_zh != null) { elem = elem_zh; }
+    
+    var canvas = document.createElement("canvas");
+    canvas.width = elem.naturalWidth;
+    canvas.height = elem.naturalHeight;
+    var ctx = canvas.getContext("2d");
+    ctx.drawImage(elem, 0, 0);
+    var dataURL = canvas.toDataURL("image/png");
+    console.log(elem.naturalHeight);
+    console.log(elem.naturalWidth);
+    elem.src = dataURL;
+}
+
+function insert_figures() {
+    insert_one_figure("fig1");
+    insert_one_figure("fig2");
+    insert_one_figure("fig3");
+    insert_one_figure("fig4");
 }

Markdown/en/about.html

@@ -1,4 +0,0 @@
-<div id="md_about" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p><b>APP</b>: This APP is developed using Gradio and deployed on HuggingFace. Due to limited resources (2 cores, 16G memory), the response may be slow. For a better experience, it is recommended to clone the source code from <a href="https://github.com/blairstar/The_Art_of_DPM" target="_blank" rel="noopener noreferrer">github</a> and run it locally. This program only relies on Gradio, SciPy, and Matplotlib.</p>
-<p><b>Author</b>: Zhenxin Zheng, Senior computer vision engineer with ten years of algorithm development experience, Formerly employed by Tencent and JD.com, currently focusing on image and video generation.</p>
-<p><b>Email</b>: <a href="mailto:blair.star@163.com" target="_blank" rel="noopener noreferrer">blair.star@163.com</a>.</p>
-</span></div></div></div>

Markdown/en/backward_process.html

@@ -1,46 +0,0 @@
-<div id="md_backward_process" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>If the final probability distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> and the posterior probabilities of each transform <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z),q(z_{t-1}|z_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> are known, the data distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> can be recovered through the Bayes Theorem and the Law of Total Probability, as shown in equations 5.1~5.4. When the final probability distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> is very similar to the standard normal distribution, the standard normal distribution can be used as a substitute.</p>
-<p>Specifics can be seen in Demo 3.2. In the example, <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> substitutes <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span>, and the error magnitude is given through JS Divergence. The restored probability distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> and <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> are identified by the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="green"><mtext>green&nbsp;curve</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{green}{\text{green curve}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord text" style="color: green;"><span class="mord" style="color: green;">green&nbsp;curve</span></span></span></span></span></span>, and the original probability distribution is identified by the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mtext>blue&nbsp;curve</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{\text{blue curve}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6944em;"></span><span class="mord text" style="color: blue;"><span class="mord" style="color: blue;">blue&nbsp;curve</span></span></span></span></span></span>. It can be observed that the data distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> can be well restored, and the error (JS Divergence) will be smaller than the error caused by the standard normal distribution replacing <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>.
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>T</mi></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>T</mi></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>…</mo></mrow></mstyle></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>t</mi></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>t</mi></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>…</mo></mrow></mstyle></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>1</mn></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(z_{T-1}) &amp;= \int q(z_{T-1},z_T)dz_T = \int q(z_{T-1}|z_T)q(z_T)dz_T                       \tag{5.1}   \newline
-               &amp; \dots                                                                              \notag      \newline
-    q(z_{t-1}) &amp;= \int q(z_{t-1},z_t)dz_t = \int q(z_{t-1}|z_t)q(z_t)dz_t                       \tag{5.2}   \newline
-               &amp; \dots                                                                              \notag      \newline
-    q(z_1)     &amp;= \int q(z_1,z_2) dz_1    = \int q(z_1|z_2)q(z_2)dz_2                           \tag{5.3}   \newline
-    q(x)       &amp;= \int q(x,z_1) dz_1      = \int q(x|z_1)q(z_1)dz_1                             \tag{5.4}   \newline
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 13.089em; vertical-align: -6.2945em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 6.7945em;"><span class="" style="top: -8.7945em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: -6.7923em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"></span></span><span class="" style="top: -4.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: -2.77em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"></span></span><span class="" style="top: -0.75em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: 1.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 6.2945em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 6.7945em;"><span class="" style="top: -8.7945em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -6.7923em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner">…</span></span></span><span class="" style="top: -4.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.77em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner">…</span></span></span><span class="" style="top: -0.75em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: 1.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 6.2945em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 6.7945em;"><span class="" style="top: -8.7945em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.1</span></span><span class="mord">)</span></span></span></span><span class="" style="top: -6.7923em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""></span></span><span class="" style="top: -4.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.2</span></span><span class="mord">)</span></span></span></span><span class="" style="top: -2.77em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""></span></span><span class="" style="top: -0.75em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.3</span></span><span class="mord">)</span></span></span></span><span class="" style="top: 1.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.4</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 6.2945em;"><span class=""></span></span></span></span></span></span></span></span></span>
-In this article, the aforementioned transform is referred to as the <b>Posterior Transform</b>. For example, in equation 5.4, the input of the transform is the probability distribution function <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>, and the output is the probability distribution function <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>.The entire transform is determined by the posterior <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>. This transform can also be considered as the linear weighted sum of a set of basis functions, where the basis functions are <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> under different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">z_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>, and the weights of each basis function are <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>. Some interesting properties of this transform will be introduced in Section 7.</p>
-<p>In Section 3, we have considered two special posterior probability distributions. Next, we analyze their corresponding <em>posterior transforms</em>.</p>
-<ul>
-    <li> When <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0</span></span></span></span></span>, the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> for different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span> are almost the same as <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>. In other words, the basis functions of linear weighted sum are almost the same. In this state, no matter how the input changes, the output of the transformation is always <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>."</li>
-    <li> When <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">1</span></span></span></span></span>, the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> for different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span> values becomes a series of Dirac delta functions and zero functions. In this state, as long as the <em>support set</em> of the input distribution is included in the <em>support set</em> of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>, the output of the transformation will remain the same with the input.</li>
-</ul>
-
-<p>In Section 5, it is mentioned that the 1000 transformations used in the DDPM<a href="#ddpm">[2]</a> can be represented using a single transformation
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>Z</mi><mi>T</mi></msub><mo>=</mo><msqrt><mn>0.0000403</mn></msqrt><mtext>&nbsp;</mtext><mi>X</mi><mo>+</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mn>0.0000403</mn></mrow></msqrt><mtext>&nbsp;</mtext><mi>ϵ</mi><mo>=</mo><mn>0.00635</mn><mtext>&nbsp;</mtext><mi>X</mi><mo>+</mo><mn>0.99998</mn><mtext>&nbsp;</mtext><mi>ϵ</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon                        \tag{5.5}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.6161em; vertical-align: -0.558em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.058em;"><span class="" style="top: -3.102em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9561em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord">0.0000403</span></span></span><span class="" style="top: -2.9161em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.1256em;"><span class=""></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord">0.00635</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord">0.99998</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">ϵ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.558em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.058em;"><span class="" style="top: -3.102em;"><span class="pstrut" style="height: 3em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.5</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.558em;"><span class=""></span></span></span></span></span></span></span></span></span></p>
-<p>Since <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>0.0000403</mn></mrow><annotation encoding="application/x-tex">\alpha=0.0000403</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0.0000403</span></span></span></span></span> is very small, the corresponding standard deviation of GaussFun (Equation 3.4) reaches 157.52. However, the range of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span> is limited within <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-1, 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span></span>, which is far smaller than the standard deviation of GaussFun. Within the range of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x \in [-1, 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5782em; vertical-align: -0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span></span>, GaussFun should be close to a constant, showing little variation. Therefore, the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> corresponding to different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">z_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span> are almost the same as <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>. In this state, the posterior transform corresponding to <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> does not depend on the input distribution, the output distribution will always be <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>.</p>
-<p><b>Therefore, theoretically, in the DDPM model, it is not necessary to use the standard normal distribution to replace <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>. Any other arbitrary distributions can also be used as a substitute.</b></p>
-</span></div></div></div>

Markdown/en/deconvolution.html

@@ -1,4 +0,0 @@
-<div id="md_deconvolution" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>As mentioned in the section 2, the transformation of Equation 2.1 can be divided into two sub-transformations, the first one being a linear transformation and the second being adding independent Gaussian noise. The linear transformation is equivalent to a scaling transform of the probability distribution, so it has an inverse transformation. Adding independent Gaussian noise is equivalent to the execution of a convolution operation on the probability distribution, which can be restored through <b>deconvolution</b>. Therefore, theoretically, the data distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> can be recovered from the final probability distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span> through <b>inverse linear transform</b> and <b>deconvolution</b>.</p>
-<p>However, in actuality, some problems do exist. Due to the extreme sensitivity of deconvolution to errors, having high input sensitivity, even a small amount of input noise can lead to significant changes in output<a href="#deconv_1">[11]</a><a href="#deconv_2">[12]</a>. Meanwhile, in the diffusion model, the standard normal distribution is used as an approximation to replace <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>, thus, noise is introduced at the initial stage of recovery. Although the noise is relatively small, because of the sensitivity of deconvolution, the noise will gradually amplify, affecting the recovery.</p>
-<p>In addition, the infeasibility of <b>deconvolution restoring</b> can be understood from another perspective. Since the process of forward transform (equations 4.1 to 4.4) is fixed, the convolution kernel is fixed. Therefore, the corresponding deconvolution transform is also fixed. Since the initial data distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> is arbitrary, any probability distribution can be transformed into an approximation of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}(0,I)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">I</span><span class="mclose">)</span></span></span></span></span> through a series of fixed linear transforms and convolutions. If <b>deconvolution restoring</b> is feasible, it means that a fixed deconvolution can be used to restore any data distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> from the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}(0,I)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">I</span><span class="mclose">)</span></span></span></span></span> , this is clearly <b>paradoxical</b>. The same input, the same transform, cannot have multiple different outputs.</p>
-</span></div></div></div>

Markdown/en/introduction.html

@@ -1,6 +0,0 @@
-<div id="md_introduction" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>The Diffusion Model<a href="#dpm">[1]</a><a href="#ddpm">[2]</a> is currently the main method used in image and video generation, but due to its abstruse theory, many engineers are unable to understand it well. This article will provide a very easy-to-understand method to help readers grasp the principles of the Diffusion Model. Specifically, it will illustrate the Diffusion Model using examples of one-dimensional random variables in an interactive way, explaining several interesting properties of the Diffusion Model in an intuitive manner.</p>
-<p>The diffusion model is a probabilistic model. Probabilistic models mainly offer two functions: calculating the probability of a given sample appearing; and generating new samples. The diffusion model focuses on the latter aspect, facilitating the production of new samples, thus realizing the task of <strong>generation</strong>.</p>
-<p>The diffusion model differs from general probability models (such as GMM), which directly models the probability distribution of random variables. The diffusion model adopts an indirect approach, which utilizes <strong>random variable transformation</strong>(shown in Figure 1a) to gradually convert the data distribution (the probability distribution to be modeled) into the <strong>standard normal distribution</strong>, and meanwhile models the posterior probability distribution corresponding to each transformation (Figure 1b-c). Upon obtaining the final standard normal distribution and the posterior probability distributions, one can generate samples of each random variable <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Z</mi><mi>T</mi></msub><mo>…</mo><msub><mi>Z</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>Z</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Z_T \ldots Z_2,Z_1,X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8778em; vertical-align: -0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span> in reverse order through Ancestral Sampling method. Simultaneously, initial data distribution <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> can be determined by employing Bayes theorem and the total probability theorem.</p>
-<p>One might wonder: indirect methods require modeling and learning T posterior probability distributions, while direct methods only need to model one probability distribution, Why would we choose the indirect approach? Here's the reasoning: the initial data distribution might be quite complex and hard to represent directly with a probability model. In contrast, the complexity of each posterior probability distribution in indirect methods is significantly simpler, allowing it to be approximated by simple probability models. As we will see later, given certain conditions, posterior probability distributions can closely resemble Gaussian distributions, thus a simple conditional Gaussian model can be used for modeling.</p>
-<center> <img style="margin-top:12px" width="820" src="file/pipe.jpg"> </center>
-<center> Figure 1: Diffusion model schematic </center></span></div></div></div>

Markdown/en/likelihood.html

@@ -1,17 +0,0 @@
-<div id="md_likelihood" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>From the transformation method (equation 1.1), it can be seen that the probability distribution of the forward conditional probability <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> is a Gaussian distribution, which is only related to the value of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>, regardless of the probability distribution of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>. 
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><msqrt><mi>α</mi></msqrt><mi>x</mi><mo separator="true">,</mo><mtext>&nbsp;</mtext><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(z|x) &amp;= \mathcal{N}(\sqrt{\alpha}x,\ 1-\alpha)    \tag{2.1}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5092em; vertical-align: -0.5046em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0046em;"><span class="" style="top: -3.1554em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5046em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0046em;"><span class="" style="top: -3.1554em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8492em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.8092em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.1908em;"><span class=""></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5046em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0046em;"><span class="" style="top: -3.1554em;"><span class="pstrut" style="height: 3em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2.1</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5046em;"><span class=""></span></span></span></span></span></span></span></span></span>
-It can be understood by concrete examples in Demo 2. The third figure depict the shape of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>. From the figure, a uniform slanting line can be observed. This implies that the mean of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> is linearly related to x, and the variance is fixed. The magnitude of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span> will determine the width and incline of the slanting line.</p>
-</span></div></div></div>

Markdown/en/posterior.html

@@ -1,122 +0,0 @@
-<div id="md_posterior" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>The posterior probability distribution does not have a closed form, but its shape can be inferred approximately through some technique.</p>
-<p>According to Bayes formula, we have
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(x|z) = \frac{q(z|x)q(x)}{q(z)}    \tag{3.1}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 2.663em; vertical-align: -1.0815em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5815em;"><span class="" style="top: -3.5815em;"><span class="pstrut" style="height: 3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.427em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.936em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.0815em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5815em;"><span class="" style="top: -3.5815em;"><span class="pstrut" style="height: 3.427em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.1</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.0815em;"><span class=""></span></span></span></span></span></span></span></span></span></p>
-<p>When <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span> takes a fixed value, <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> is a constant, so the shape of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> is only related to <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{q(z|x)q(x)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></span>.
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo><mo>∝</mo><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mtext>&nbsp;</mtext><mi>z</mi><mtext>&nbsp;</mtext><mi>i</mi><mi>s</mi><mtext>&nbsp;</mtext><mi>f</mi><mi>i</mi><mi>x</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(x|z)  \propto q(z|x)q(x)      \qquad where\ z\ is\ fixed  \tag{3.2}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5em; vertical-align: -0.5em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1em;"><span class="" style="top: -3.16em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right: 2em;"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ere</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span><span class="mord mathnormal">i</span><span class="mord mathnormal">x</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1em;"><span class="" style="top: -3.16em;"><span class="pstrut" style="height: 3em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.2</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5em;"><span class=""></span></span></span></span></span></span></span></span></span></p>
-<p>From Equation 2.1, we can see that <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> is a Gaussian distribution, so we have
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∝</mo><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mi>π</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></msqrt></mfrac><mi>exp</mi><mo>⁡</mo><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><msqrt><mi>α</mi></msqrt><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><mtext>&nbsp;</mtext><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mspace width="2em"></mspace></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mtext>&nbsp;</mtext><mi>z</mi><mtext>&nbsp;</mtext><mi>i</mi><mi>s</mi><mtext>&nbsp;</mtext><mi>f</mi><mi>i</mi><mi>x</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><msqrt><mi>α</mi></msqrt></mfrac><munder><munder><mrow><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><mi>exp</mi><mo>⁡</mo><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo stretchy="true">⏟</mo></munder><mtext>GaussFun</mtext></munder><mtext>&nbsp;</mtext><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mspace width="2em"></mspace></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mtext>&nbsp;</mtext><mi>μ</mi><mo>=</mo><mfrac><mi>z</mi><msqrt><mi>α</mi></msqrt></mfrac><mspace width="1em"></mspace><mi>σ</mi><mo>=</mo><msqrt><mfrac><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><mi>α</mi></mfrac></msqrt></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(x|z)  &amp;\propto \frac{1}{\sqrt{2\pi(1-\alpha)}}\exp{\frac{-(z-\sqrt{\alpha}x)^2}{2(1-\alpha)}}\ q(x)&amp;  \qquad &amp;where\ z\ is\ fixed      \tag{3.3}   \newline
-            &amp;=      \frac{1}{\sqrt{\alpha}} \underbrace{\frac{1}{\sqrt{2\pi}\sigma}\exp{\frac{-(x-\mu)^2}{2\sigma^2}}}_{\text{GaussFun}}\ q(x)&amp; \qquad &amp;where\ \mu=\frac{z}{\sqrt{\alpha}}\quad \sigma=\sqrt{\frac{1-\alpha}{\alpha}}   \tag{3.4}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 7.129em; vertical-align: -3.3145em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3214em;"><span class="" style="top: -2.175em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.935em;"><span class="svg-align" style="top: -3.2em;"><span class="pstrut" style="height: 3.2em;"></span><span class="mord" style="padding-left: 1em;"><span class="mord">2</span><span class="mord mathnormal" style="margin-right: 0.0359em;">π</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mclose">)</span></span></span><span class="" style="top: -2.895em;"><span class="pstrut" style="height: 3.2em;"></span><span class="hide-tail" style="min-width: 1.02em; height: 1.28em;"><svg width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2397em;"><span class=""></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.936em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3214em;"><span class="" style="top: -2.3097em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8003em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7603em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal">μ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.578em;"><span class=""></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 2.2563em;"><span class=""></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mspace" style="margin-right: 2em;"></span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mspace" style="margin-right: 2em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ere</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span><span class="mord mathnormal">i</span><span class="mord mathnormal">x</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ere</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.1076em;"><span class="" style="top: -2.3097em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8003em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7603em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.7884em;"><span class=""></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.3</span></span><span class="mord">)</span></span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.4</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span></span></span></span></span></p>
-<p>It can be observed that the <b>GaussFun</b> part is a Gaussian function of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span>, with a mean of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>z</mi><msqrt><mi>α</mi></msqrt></mfrac></mrow><annotation encoding="application/x-tex">\frac{z}{\sqrt{\alpha}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.2334em; vertical-align: -0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.6954em;"><span class="" style="top: -2.6259em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8059em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mtight" style="padding-left: 0.833em;"><span class="mord mathnormal mtight" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7659em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail mtight" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2341em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.538em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span> and a variance of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mfrac><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><mi>α</mi></mfrac></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\frac{1-\alpha}{\alpha}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.84em; vertical-align: -0.6049em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.2351em;"><span class="svg-align" style="top: -3.8em;"><span class="pstrut" style="height: 3.8em;"></span><span class="mord" style="padding-left: 1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8451em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0037em;">α</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right: 0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span class="" style="top: -3.1951em;"><span class="pstrut" style="height: 3.8em;"></span><span class="hide-tail" style="min-width: 1.02em; height: 1.88em;"><svg width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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-M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.6049em;"><span class=""></span></span></span></span></span></span></span></span></span>, so the shape of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> is determined by <strong>the product of GaussFun and q(x)</strong>.</p>
-<p>According to the characteristics of <em>multiplication</em>, the characteristics of the shape of the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> function can be summarized. </p>
-<ul>
-<li>When the variance of the Gaussian function is small (small noise), or when <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> changes slowly, the shape of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> will approximate to the Gaussian function, and have a simpler function form, which is convenient for modeling and learning.</li>
- 
-<li>When the variance of the Gaussian function is large (large noise), or when <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> changes drastically, the shape of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> will be more complex, and greatly differ from a Gaussian function, which makes it difficult to model and learn.</li>
-</ul>
-
-<p>The specifics can be seen in Demo 2. The fourth figure present the shape of the posterior <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>, which shows an irregular shape and resembles a curved and uneven line. As <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span> increases (noise decreases), the curve tends to be uniform and straight. Readers can adjust different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span> values and observe the relationship between the shape of posterior and the level of noise. In the last figure, the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mtext>blue&nbsp;dash&nbsp;line</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{\text{blue dash line}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6944em;"></span><span class="mord text" style="color: blue;"><span class="mord" style="color: blue;">blue&nbsp;dash&nbsp;line</span></span></span></span></span></span> represents <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>, the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="green"><mtext>green&nbsp;dash&nbsp;line</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{green}{\text{green dash line}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8889em; vertical-align: -0.1944em;"></span><span class="mord text" style="color: green;"><span class="mord" style="color: green;">green&nbsp;dash&nbsp;line</span></span></span></span></span></span> represents <b>GaussFun</b> in the equation 3.4, and the <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="orange"><mtext>orange&nbsp;curve</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{orange}{\text{orange curve}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.625em; vertical-align: -0.1944em;"></span><span class="mord text" style="color: orange;"><span class="mord" style="color: orange;">orange&nbsp;curve</span></span></span></span></span></span> represents the result of multiplying the two function and normalizing it, which is the posterior probability <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo>=</mo><mi>f</mi><mi>i</mi><mi>x</mi><mi>e</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z=fixed)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span><span class="mord mathnormal">i</span><span class="mord mathnormal">x</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span></span> under a fixed z condition. Readers can adjust different values of z to observe how the fluctuation of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> affect the shape of the posterior probability <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>.</p>
-<p>The posterior <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> under two special states are worth considering.</p>
-<ul>
-<li>As <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0</span></span></span></span></span>, the variance of <b>GaussFun</b> tends to <b><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord">∞</span></span></span></span></span></b>, and <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> for different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span> almost become identical, and almost the same as <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>. Readers can set <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span> to 0.01 in Demo 2 to observe the specific results.</li>
-    
-<li>As <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">1</span></span></span></span></span>, the variance of <b>GaussFun</b> tends to <b><span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0</span></span></span></span></span></b>, The <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> for different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span> values contract into a series of <em>Dirac delta functions</em> with different offsets equalling to <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span>. However, there are some exceptions. When there are regions where <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span> is zero, the corresponding <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span> will no longer be a Dirac <em>delta function</em>, but a zero function. Readers can set <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span> to 0.999 in Demo 2 to observe the specific results.</li>
-</ul></span></div></div></div>

Markdown/en/reference.html

@@ -1,16 +0,0 @@
-<div id="md_reference" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p><a href="https://arxiv.org/abs/1503.03585" id="dpm" target="_blank" rel="noopener noreferrer"> [1] Deep Unsupervised Learning Using Nonequilibrium Thermodynami </a></p>
-<p><a href="https://arxiv.org/abs/1503.03585" id="ddpm" target="_blank" rel="noopener noreferrer"> [2] Denoising Diffusion Probabilistic Models </a></p>
-<p><a href="https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables" id="linear_transform" target="_blank" rel="noopener noreferrer"> [3] Linear Transformations of Random Variable </a></p>
-<p><a href="https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables" id="sum_conv" target="_blank" rel="noopener noreferrer"> [4] Sums and Convolution </a></p>
-<p><a href="https://en.wikipedia.org/wiki/Banach_fixed-point_theorem" id="fixed_point" target="_blank" rel="noopener noreferrer"> [5] Banach fixed-point theorem </a></p>
-<p><a href="https://en.wikipedia.org/wiki/Contraction_mapping" id="ctr" target="_blank" rel="noopener noreferrer"> [6] Contraction mapping </a></p>
-<p><a href="https://stats.libretexts.org/Bookshelves/Probability_Theory/Book%3A_Introductory_Probability_(Grinstead_and_Snell)/11%3A_Markov_Chains/11.04%3A_Fundamental_Limit_Theorem_for_Regular_Chains" id="mc_limit" target="_blank" rel="noopener noreferrer"> [7] Fundamental Limit Theorem for Regular Chains </a></p>
-<p><a href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf" id="mc_basic_p6" target="_blank" rel="noopener noreferrer"> [8] Markov Chain:Basic Theory - Proposition 6 </a></p>
-<p><a href="https://arxiv.org/abs/1702.07339" id="fp_converse" target="_blank" rel="noopener noreferrer"> [9] A Converse to Banach's Fixed Point Theorem and its CLS Completeness </a></p>
-<p><a href="https://en.wikipedia.org/wiki/Cross-entropy#Cross-entropy_minimization" id="ce_kl" target="_blank" rel="noopener noreferrer"> [10] Cross-entropy minimization </a></p>
-<p><a href="https://thewolfsound.com/deconvolution-inverse-convolution/" id="deconv_1" target="_blank" rel="noopener noreferrer"> [11] Deconvolution Using Frequency-Domain Division </a></p>
-<p><a href="https://www.strollswithmydog.com/deconvolution-by-division-in-the-frequency-domain/" id="deconv_2" target="_blank" rel="noopener noreferrer"> [12] deconvolution-by-division-in-the-frequency-domain </a></p>
-<p><a href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf" id="mc_basic_t7" target="_blank" rel="noopener noreferrer"> [13] Markov Chain:Basic Theory - Theorem 7 </a></p>
-<p><a href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf" id="mc_basic_d4" target="_blank" rel="noopener noreferrer"> [14] Markov Chain:Basic Theory - Definition 4 </a>
- </p>
-</span></div></div></div>

Markdown/en/transform.html

@@ -1,66 +0,0 @@
-<div id="md_transform" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>To transform the initial data distribution into a simple standard normal distribution, the diffusion model uses the following transformation method: 
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>=</mo><msqrt><mi>α</mi></msqrt><mi>X</mi><mo>+</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msqrt><mi>ϵ</mi><mspace width="2em"></mspace><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mspace width="1em"></mspace><mi>α</mi><mo>&lt;</mo><mn>1</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>ϵ</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    Z = \sqrt{\alpha} X +  \sqrt{1-\alpha}\epsilon \qquad where \quad \alpha &lt; 1, \quad \epsilon \sim \mathcal{N}(0, I)  \tag{1.1}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5744em; vertical-align: -0.5372em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0372em;"><span class="" style="top: -3.1228em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8492em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.8092em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.1256em;"><span class=""></span></span></span></span></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right: 2em;"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ere</span><span class="mspace" style="margin-right: 1em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 1em;"></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">I</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5372em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0372em;"><span class="" style="top: -3.1228em;"><span class="pstrut" style="height: 3em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1.1</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5372em;"><span class=""></span></span></span></span></span></span></span></span></span>
-where <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\sim q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>is any random variable，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo>∼</mo><mi>q</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z\sim q(Z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="mclose">)</span></span></span></span></span> is the transformed random variable。</p>
-<p>This transformation can be divided into two sub-transformations。</p>
-<p>The first sub-transformation performs a linear transformation (<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mi>α</mi></msqrt><mi>X</mi></mrow><annotation encoding="application/x-tex">\sqrt{\alpha}X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04em; vertical-align: -0.2397em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8003em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7603em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
-c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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-c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
-s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
-c69,-144,104.5,-217.7,106.5,-221
-l0 -0
-c5.3,-9.3,12,-14,20,-14
-H400000v40H845.2724
-s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
-c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2397em;"><span class=""></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span>) on the random variable <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span>. According to the conclusion of the literature<a href="#linear_transform">[3]</a>, the linear transformation makes the probability distribution of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span> <strong>narrower and taller</strong>, and the extent of <strong>narrowing and heightening</strong> is directly proportional to the value of <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>. </p>
-<p>This can be specifically seen in Demo 1, where the first figure depicts a randomly generated one-dimensional probability distribution, and the second figure represents the probability distribution after the linear transformation. It can be observed that the curve of the third figure has become <strong>narrower and taller</strong> compared to the first image. Readers can experiment with different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span> to gain a more intuitive understanding.</p>
-<p>The second sub-transformation is <strong>adding independent random noise</strong>(<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msqrt><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\sqrt{1-\alpha}\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04em; vertical-align: -0.1744em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8656em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.8256em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
-c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
-c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
-c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
-s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
-c69,-144,104.5,-217.7,106.5,-221
-l0 -0
-c5.3,-9.3,12,-14,20,-14
-H400000v40H845.2724
-s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
-c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.1744em;"><span class=""></span></span></span></span></span><span class="mord mathnormal">ϵ</span></span></span></span></span>). According to the conclusion of the literature<a href="#sum_conv">[4]</a>, <strong>adding independent random variables</strong> is equivalent to performing convolution on the two probability distributions. Since the probability distribution of random noise is Gaussian, it is equivalent to performing a <strong>Gaussian Blur</strong> operation. After blurring, the original probability distribution will become smoother and more similar to the standard normal distribution. The degree of blurring is directly proportional to the noise level (<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{1-\alpha}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04em; vertical-align: -0.1744em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8656em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.8256em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
-c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
-c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
-c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
-s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
-c69,-144,104.5,-217.7,106.5,-221
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-s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
-c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.1744em;"><span class=""></span></span></span></span></span></span></span></span></span>).</p>
-<p>For specifics, one can see Demo 1, where the first figure is a randomly generated one-dimensional probability distribution, and the third  figure is the result after the transformation. It can be seen that the transformed probability distribution curve is smoother and there are fewer corners. The readers can test different <span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span> values to feel how the noise level affect the shape of the probability distribution. The last figure is the result after applying all two sub-transformations.</p>
-</span></div></div></div>

Markdown/zh/about.html

@@ -1,4 +0,0 @@
-<div id="md_about" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p><b>APP</b>: 本APP是使用Gradio开发，并部署在HuggingFace。由于资源有限(2核，16G内存)，所以可能会响应较慢。为了更好地体验，建议从<a href="https://github.com/blairstar/The_Art_of_DPM" target="_blank" rel="noopener noreferrer">github</a>复制源代码，在本地机器运行。本APP只依赖Gradio, SciPy, Matplotlib。</p>
-<p><b>Author</b>: 郑镇鑫，资深视觉算法工程师，十年算法开发经历，曾就职于腾讯京东等互联网公司，目前专注于视频生成(类似Sora)。</p>
-<p><b>Email</b>: <a href="mailto:blair.star@163.com" target="_blank" rel="noopener noreferrer">blair.star@163.com</a> 。</p>
-</span></div></div></div>

Markdown/zh/backward_process.html

@@ -1,47 +0,0 @@
-<div id="md_backward_process" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>如果知道了最终的概率分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>及各个转换过程的后验概率<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z),q(z_{t-1}|z_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>，则可通过“贝叶斯公式”和“全概率公式”恢复数据分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>，见式5.1~5.4。当最终的概率分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>与标准正态分布很相似时，可用标准正态分布代替。</p>
-<p>具体可看Demo 3.2。示例中<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>使用<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span>代替，同时通过JS Div给出了误差大小。恢复的概率分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>及<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>使用<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="green"><mtext>绿色曲线</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{green}{绿色曲线}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback" style="color: green;">绿色曲线</span></span></span></span></span>标识，原始的概率分布使用<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mtext>蓝色曲线</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{蓝色曲线}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback" style="color: blue;">蓝色曲线</span></span></span></span></span>标识。可以看出，数据分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>能够被很好地恢复回来，并且误差(JS Divergence)会小于标准正态分布替换<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>引起的误差。
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>T</mi></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>T</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>T</mi></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>…</mo></mrow></mstyle></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>t</mi></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mi>t</mi></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>…</mo></mrow></mstyle></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>2</mn></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mo>∫</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>d</mi><msub><mi>z</mi><mn>1</mn></msub></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(z_{T-1}) &amp;= \int q(z_{T-1},z_T)dz_T = \int q(z_{T-1}|z_T)q(z_T)dz_T                       \tag{5.1}   \newline
-               &amp; \dots                                                                              \notag      \newline
-    q(z_{t-1}) &amp;= \int q(z_{t-1},z_t)dz_t = \int q(z_{t-1}|z_t)q(z_t)dz_t                       \tag{5.2}   \newline
-               &amp; \dots                                                                              \notag      \newline
-    q(z_1)     &amp;= \int q(z_1,z_2) dz_1    = \int q(z_1|z_2)q(z_2)dz_2                           \tag{5.3}   \newline
-    q(x)       &amp;= \int q(x,z_1) dz_1      = \int q(x|z_1)q(z_1)dz_1                             \tag{5.4}   \newline
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 13.089em; vertical-align: -6.2945em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 6.7945em;"><span class="" style="top: -8.7945em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: -6.7923em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"></span></span><span class="" style="top: -4.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: -2.77em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"></span></span><span class="" style="top: -0.75em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span><span class="" style="top: 1.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 6.2945em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 6.7945em;"><span class="" style="top: -8.7945em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -6.7923em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner">…</span></span></span><span class="" style="top: -4.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2083em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.2806em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -2.77em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner">…</span></span></span><span class="" style="top: -0.75em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: 1.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mop op-symbol large-op" style="margin-right: 0.4445em; position: relative; top: -0.0011em;">∫</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 6.2945em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 6.7945em;"><span class="" style="top: -8.7945em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.1</span></span><span class="mord">)</span></span></span></span><span class="" style="top: -6.7923em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""></span></span><span class="" style="top: -4.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.2</span></span><span class="mord">)</span></span></span></span><span class="" style="top: -2.77em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""></span></span><span class="" style="top: -0.75em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.3</span></span><span class="mord">)</span></span></span></span><span class="" style="top: 1.7722em;"><span class="pstrut" style="height: 3.36em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">5.4</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 6.2945em;"><span class=""></span></span></span></span></span></span></span></span></span>
-在本文中，将上述恢复过程(式5.1~5.4)所使用的变换称之为“后验概率变换”。例如，在式5.4中，变换的输入为概率分布函数<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>，输出为概率分布函数<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>，整个变换由后验概率分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>决定。此变换也可看作为一组基函数的线性加权和，基函数为不同条件下的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>，各个基函数的权重为<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>。在第7节，将会进一步介绍此变换的一些有趣性质。</p>
-<p>在第3节中，我们考虑了两个特殊的后验概率分布。接下来，分析其对应的”后验概率变换“。</p>
-<ul>
-    <li> 当<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0</span></span></span></span></span>时，不同<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span>值的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>均与<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>几乎相同，也就是说，线性加权和的基函数几乎相同。此状态下，不管输入如何变化，变换的输出总为<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>。</li>
-    <li> 当<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">1</span></span></span></span></span>时，不同<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span>值的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>收缩成一系列不同偏移量的Dirac delta函数及零函数。此状态下，只要输入分布的支撑集(support set)包含于<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>的支撑集，变换的输出与输入将保持一致。</li>
-</ul>
-
-<p>在第5节中提到，DDPM<a href="#ddpm">[2]</a>论文所使用的1000次变换可使用一次变换表示：
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>Z</mi><mi>T</mi></msub><mo>=</mo><msqrt><mn>0.0000403</mn></msqrt><mtext>&nbsp;</mtext><mi>X</mi><mo>+</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mn>0.0000403</mn></mrow></msqrt><mtext>&nbsp;</mtext><mi>ϵ</mi><mo>=</mo><mn>0.00635</mn><mtext>&nbsp;</mtext><mi>X</mi><mo>+</mo><mn>0.99998</mn><mtext>&nbsp;</mtext><mi>ϵ</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon                        \tag{5.5}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.6161em; vertical-align: -0.558em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.058em;"><span class="" style="top: -3.102em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.9561em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord">0.0000403</span></span></span><span class="" style="top: -2.9161em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-由于<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mn>0.0000403</mn></mrow><annotation encoding="application/x-tex">\alpha=0.0000403</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0.0000403</span></span></span></span></span>非常小，其对应的GaussFun(式3.4)的标准差达到157.52，而<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span>的范围限制在<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-1, 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span></span>，远小于GaussFun的标准差。在<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x \in [-1, 1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5782em; vertical-align: -0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span></span>范围内，GaussFun应该接近于常量，没有什么变化，所以不同的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>z</mi><mi>T</mi></msub></mrow><annotation encoding="application/x-tex">z_T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.5806em; vertical-align: -0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span></span></span></span></span>对应的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>均与<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>几乎相同。在这种状态下，对于<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>相应的后验概率变换，不管输入分布是什么，输出分布都将是<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>。</p>
-<p><b>所以，理论上，在DDPM模型中，无需非得使用标准正态分布代替<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>，也可使用其它任意的分布代替。</b>
- </p>
-</span></div></div></div>

Markdown/zh/deconvolution.html

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-<div id="md_deconvolution" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>在第二节中提到，式2.1的变换可分为两个子变换，第一个子变换为”线性变换“，第二个为“加上独立高斯噪声”。线性变换相当于对概率分布进行拉伸变换，所以存在逆变换。"加上独立高斯噪声”相当于对概率分布执行卷积操作，卷积操作可通过逆卷积恢复。所以，理论上，可通过“逆线性变换”和“逆卷积”从最终的概率分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>恢复数据分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>。</p>
-<p>但实际上，会存在一些问题。由于逆卷积对误差极为敏感，具有很高的输入灵敏度，很小的输入噪声就会引起输出极大的变化<a href="#deconv_1">[11]</a><a href="#deconv_2">[12]</a>。而在扩散模型中，会使用标准正态分布近似代替<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><msub><mi>z</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z_T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.044em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>，因此，在恢复的起始阶段就会引入噪声。虽然噪声较小，但由于逆卷积的敏感性，噪声会逐步放大，影响恢复。</p>
-<p>另外，也可以从另一个角度理解“逆卷积恢复”的不可行性。由于前向变换的过程(式4.1~4.4)是确定的，所以卷积核是固定的，因此，相应的“逆卷积变换“也是固定的。由于起始的数据分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>可以是任意的分布，所以，通过一系列固定的“卷积正变换”，可以将任意的概率分布转换成近似<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}(0,I)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">I</span><span class="mclose">)</span></span></span></span></span>的分布。如“逆卷积变换“可行，则意味着，可用一个固定的“逆卷积变换"，将<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}(0,I)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">I</span><span class="mclose">)</span></span></span></span></span>分布恢复成任意的数据分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>，这明显是一个悖论。同一个输入，同一个变换，不可能会有多个输出。</p>
-</span></div></div></div>

Markdown/zh/introduction.html

@@ -1,6 +0,0 @@
-<div id="md_introduction" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>扩散模型<a href="#dpm">[1]</a><a href="#ddpm">[2]</a>是当前图像生成和视频生成使用的主要方式，但由于其晦涩的理论，很多工程师并不能很好地理解。本文将提供一种非常直观易懂的方式，方便读者理解把握扩散模型的原理。特别地，将以互动的形式，以一维随机随机变量的扩散模型进行举例，直观解释扩散模型的多个有趣的性质。</p>
-<p>扩散模型是一个概率模型。概率模型主要提供两方面的功能：计算给定样本出现的概率；采样生成新样本。扩散模型侧重于第二方面，方便采样新样本，从而实现"生成"的任务。</p>
-<p>扩散模型与一般的概率模型(如GMM)不同，直接建模随机变量的概率分布。扩散模型采用一种间接方式，利用“随机变量变换”的方式(如图1a)，逐步将待建模的概率分布(数据分布)转变成"标准正态分布"，同时，建模学习各个变换对应的后验概率分布(图1b-c)。有了最终的标准正态分布和各个后验概率分布，则可通过祖先采样的方式，从反向逐步采样得到各个随机变量<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Z</mi><mi>T</mi></msub><mo>…</mo><msub><mi>Z</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>Z</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Z_T \ldots Z_2,Z_1,X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.8778em; vertical-align: -0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3283em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right: 0.1389em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="minner">…</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.3011em;"><span class="" style="top: -2.55em; margin-left: -0.0715em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.15em;"><span class=""></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span>的样本。同时也可通过贝叶斯公式和全概率公式确定初始的数据分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>。</p>
-<p>可能会有这样的疑问：间接的方式需要建模学习T个后验概率分布，直接方式只需要建模学习一个概率分布，为什么要选择间接的方式呢？是这样子的：初始的数据分布可能很复杂，很难用一个概率模型直接表示；而对于间接的方式，各个后验概率分布的复杂度会简单许多，可以用简单的概率模型进行拟合。下面将会看到，当满足一些条件时，后验概率分布将非常接近高斯分布，所以可以使用简单的条件高斯模型进行建模。</p>
-<center> <img style="margin-top:12px" width="820" src="file/pipe.jpg"> </center>
-<center> Figure 1: Diffusion model schematic </center></span></div></div></div>

Markdown/zh/likelihood.html

@@ -1,17 +0,0 @@
-<div id="md_likelihood" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>由变换的方式(式1.1)可以看出，前向条件概率<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>的概率分布为高斯分布，且只与<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>的值有关，与<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>的概率分布无关。
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><msqrt><mi>α</mi></msqrt><mi>x</mi><mo separator="true">,</mo><mtext>&nbsp;</mtext><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(z|x) &amp;= \mathcal{N}(\sqrt{\alpha}x,\ 1-\alpha)    \tag{2.1}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5092em; vertical-align: -0.5046em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0046em;"><span class="" style="top: -3.1554em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5046em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0046em;"><span class="" style="top: -3.1554em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathcal" style="margin-right: 0.1474em;">N</span><span class="mopen">(</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8492em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.8092em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.1908em;"><span class=""></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right: 0.1667em;"></span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5046em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0046em;"><span class="" style="top: -3.1554em;"><span class="pstrut" style="height: 3em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2.1</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5046em;"><span class=""></span></span></span></span></span></span></span></span></span>
-具体可看Demo 2，左3图展示了<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>的形状，从图中可以看到一条均匀的斜线，这意味着<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>的均值与x线性相关，方差固定不变。<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>值的大小将决定斜线宽度和倾斜程度。</p>
-</span></div></div></div>

Markdown/zh/posterior.html

@@ -1,120 +0,0 @@
-<div id="md_posterior" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>后验概率分布没有闭合的形式，但可以通过一些方法，推断其大概的形状，并分析影响其形状的因素。</p>
-<p>根据Bayes公式，有
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(x|z) = \frac{q(z|x)q(x)}{q(z)}    \tag{3.1}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 2.663em; vertical-align: -1.0815em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5815em;"><span class="" style="top: -3.5815em;"><span class="pstrut" style="height: 3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.427em;"><span class="" style="top: -2.314em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.936em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.0815em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.5815em;"><span class="" style="top: -3.5815em;"><span class="pstrut" style="height: 3.427em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.1</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.0815em;"><span class=""></span></span></span></span></span></span></span></span></span></p>
-<p>当<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span>是取固定值时，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>是常数，所以<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>的形状只与<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{q(z|x)q(x)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></span>有关。
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo><mo>∝</mo><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mtext>&nbsp;</mtext><mi>z</mi><mtext>&nbsp;</mtext><mi>i</mi><mi>s</mi><mtext>&nbsp;</mtext><mi>f</mi><mi>i</mi><mi>x</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(x|z)  \propto q(z|x)q(x)      \qquad where\ z\ is\ fixed  \tag{3.2}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5em; vertical-align: -0.5em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1em;"><span class="" style="top: -3.16em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right: 2em;"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ere</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span><span class="mord mathnormal">i</span><span class="mord mathnormal">x</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1em;"><span class="" style="top: -3.16em;"><span class="pstrut" style="height: 3em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.2</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.5em;"><span class=""></span></span></span></span></span></span></span></span></span>
-由式2.1可知，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(z|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>为高斯分布，于是有
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∝</mo><mfrac><mn>1</mn><msqrt><mrow><mn>2</mn><mi>π</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></msqrt></mfrac><mi>exp</mi><mo>⁡</mo><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><msqrt><mi>α</mi></msqrt><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mfrac><mtext>&nbsp;</mtext><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mspace width="2em"></mspace></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mtext>&nbsp;</mtext><mi>z</mi><mtext>&nbsp;</mtext><mi>i</mi><mi>s</mi><mtext>&nbsp;</mtext><mi>f</mi><mi>i</mi><mi>x</mi><mi>e</mi><mi>d</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><msqrt><mi>α</mi></msqrt></mfrac><munder><munder><mrow><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><mi>exp</mi><mo>⁡</mo><mfrac><mrow><mo>−</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo stretchy="true">⏟</mo></munder><mtext>GaussFun</mtext></munder><mtext>&nbsp;</mtext><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mspace width="2em"></mspace></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mtext>&nbsp;</mtext><mi>μ</mi><mo>=</mo><mfrac><mi>z</mi><msqrt><mi>α</mi></msqrt></mfrac><mspace width="1em"></mspace><mi>σ</mi><mo>=</mo><msqrt><mfrac><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><mi>α</mi></mfrac></msqrt></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    q(x|z)  &amp;\propto \frac{1}{\sqrt{2\pi(1-\alpha)}}\exp{\frac{-(z-\sqrt{\alpha}x)^2}{2(1-\alpha)}}\ q(x)&amp;  \qquad &amp;where\ z\ is\ fixed      \tag{3.3}   \newline
-            &amp;=      \frac{1}{\sqrt{\alpha}} \underbrace{\frac{1}{\sqrt{2\pi}\sigma}\exp{\frac{-(x-\mu)^2}{2\sigma^2}}}_{\text{GaussFun}}\ q(x)&amp; \qquad &amp;where\ \mu=\frac{z}{\sqrt{\alpha}}\quad \sigma=\sqrt{\frac{1-\alpha}{\alpha}}   \tag{3.4}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 7.129em; vertical-align: -3.3145em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∝</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3214em;"><span class="" style="top: -2.175em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.935em;"><span class="svg-align" style="top: -3.2em;"><span class="pstrut" style="height: 3.2em;"></span><span class="mord" style="padding-left: 1em;"><span class="mord">2</span><span class="mord mathnormal" style="margin-right: 0.0359em;">π</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mclose">)</span></span></span><span class="" style="top: -2.895em;"><span class="pstrut" style="height: 3.2em;"></span><span class="hide-tail" style="min-width: 1.02em; height: 1.28em;"><svg width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2397em;"><span class=""></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.936em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.3214em;"><span class="" style="top: -2.3097em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8003em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7603em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.677em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord">−</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal">μ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span class="" style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.686em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 1.578em;"><span class=""></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 2.2563em;"><span class=""></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span><span class="arraycolsep" style="width: 1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mspace" style="margin-right: 2em;"></span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mspace" style="margin-right: 2em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ere</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span><span class="mord mathnormal">i</span><span class="mord mathnormal">x</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right: 0.0269em;">w</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ere</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.1076em;"><span class="" style="top: -2.3097em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8003em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7603em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.7884em;"><span class=""></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 3.8145em;"><span class="" style="top: -5.975em;"><span class="pstrut" style="height: 3.6516em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.3</span></span><span class="mord">)</span></span></span></span><span class="" style="top: -2.8934em;"><span class="pstrut" style="height: 3.6516em;"></span><span class=""><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">3.4</span></span><span class="mord">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 3.3145em;"><span class=""></span></span></span></span></span></span></span></span></span></p>
-<p>可以看出，<b>GaussFun</b>部分是关于<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span></span>的高斯函数，均值为<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>z</mi><msqrt><mi>α</mi></msqrt></mfrac></mrow><annotation encoding="application/x-tex">\frac{z}{\sqrt{\alpha}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.2334em; vertical-align: -0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.6954em;"><span class="" style="top: -2.6259em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8059em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord mtight" style="padding-left: 0.833em;"><span class="mord mathnormal mtight" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7659em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail mtight" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2341em;"><span class=""></span></span></span></span></span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.044em;">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.538em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>，方差为<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mfrac><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><mi>α</mi></mfrac></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\frac{1-\alpha}{\alpha}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.84em; vertical-align: -0.6049em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.2351em;"><span class="svg-align" style="top: -3.8em;"><span class="pstrut" style="height: 3.8em;"></span><span class="mord" style="padding-left: 1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8451em;"><span class="" style="top: -2.655em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right: 0.0037em;">α</span></span></span></span><span class="" style="top: -3.23em;"><span class="pstrut" style="height: 3em;"></span><span class="frac-line" style="border-bottom-width: 0.04em;"></span></span><span class="" style="top: -3.394em;"><span class="pstrut" style="height: 3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right: 0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.345em;"><span class=""></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span class="" style="top: -3.1951em;"><span class="pstrut" style="height: 3.8em;"></span><span class="hide-tail" style="min-width: 1.02em; height: 1.88em;"><svg width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
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-<p>根据”乘法“的特点，可以总结<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>函数形状具有的特点。</p>
-<ul>
-    <li> 当高斯函数的方差较小(较小噪声)，或者<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>变化缓慢时，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>的形状将近似于高斯函数，函数形式较简单，方便建模学习。</li>
-    <li> 当高斯函数的方差较大(较大噪声)，或者<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>剧烈变化时，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>的形状将较复杂，与高斯函数有较大的差别，难以建模学习。</li>
-</ul>
-
-<p>具体可看Demo 2，左4图给出后验概率分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>的形态，可以看出，其形状较不规则，像一条弯曲且不均匀的曲线。当<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>较大时(噪声较小)，曲线将趋向于均匀且笔直。读者可调整不同的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>值，观察后验概率分布与噪声大小的关系；左5图，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mtext>蓝色虚线</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{蓝色虚线}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback" style="color: blue;">蓝色虚线</span></span></span></span></span>给出<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="green"><mtext>绿色虚线</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{green}{绿色虚线}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback" style="color: green;">绿色虚线</span></span></span></span></span>给出式3.4中的GaussFun，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="orange"><mtext>黄色实线</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{orange}{黄色实线}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord cjk_fallback" style="color: orange;">黄色实线</span></span></span></span></span>给出两者相乘并归一化的结果，即固定z条件下后验概率<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo>=</mo><mi>f</mi><mi>i</mi><mi>x</mi><mi>e</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z=fixed)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.1076em;">f</span><span class="mord mathnormal">i</span><span class="mord mathnormal">x</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mclose">)</span></span></span></span></span>。读者可调整不同z值，观察<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>的波动变化对后验概率<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>形态的影响。</p>
-<p>两个特殊状态下的后验概率分布<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>值得考虑一下。</p>
-<ul>
-    <li> 当<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">0</span></span></span></span></span>时，GaussFun的方差趋向于<b>无穷大</b>，不同<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span>值的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>几乎变成一致，并与<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>几乎相同。读者可在Demo 2中，将<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>设置为0.01，观察具体的结果。</li>
-    <li> 当<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 0.6444em;"></span><span class="mord">1</span></span></span></span></span>时，GaussFun的方差趋向于<b>无穷小</b>，不同<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span>值的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(x|z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span><span class="mclose">)</span></span></span></span></span>收缩成一系列不同偏移量的Dirac delta函数, 偏移量等于<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.044em;">z</span></span></span></span></span>。但有一些例外，当q(x)存在为零的区域时，其对应的q(x|z)将不再为Dirac delta函数，而是零函数。可在Demo 2中，将<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>设置为0.999，观察具体的结果。</li>
-</ul></span></div></div></div>

Markdown/zh/reference.html

@@ -1,15 +0,0 @@
-<div id="md_reference" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p><a href="https://arxiv.org/abs/1503.03585" id="dpm" target="_blank" rel="noopener noreferrer"> [1] Deep Unsupervised Learning Using Nonequilibrium Thermodynami </a></p>
-<p><a href="https://arxiv.org/abs/1503.03585" id="ddpm" target="_blank" rel="noopener noreferrer"> [2] Denoising Diffusion Probabilistic Models </a></p>
-<p><a href="https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables" id="linear_transform" target="_blank" rel="noopener noreferrer"> [3] Linear Transformations of Random Variable </a></p>
-<p><a href="https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables" id="sum_conv" target="_blank" rel="noopener noreferrer"> [4] Sums and Convolution </a></p>
-<p><a href="https://en.wikipedia.org/wiki/Banach_fixed-point_theorem" id="fixed_point" target="_blank" rel="noopener noreferrer"> [5] Banach fixed-point theorem </a></p>
-<p><a href="https://en.wikipedia.org/wiki/Contraction_mapping" id="ctr" target="_blank" rel="noopener noreferrer"> [6] Contraction mapping </a></p>
-<p><a href="https://stats.libretexts.org/Bookshelves/Probability_Theory/Book%3A_Introductory_Probability_(Grinstead_and_Snell)/11%3A_Markov_Chains/11.04%3A_Fundamental_Limit_Theorem_for_Regular_Chains" id="mc_limit" target="_blank" rel="noopener noreferrer"> [7] Fundamental Limit Theorem for Regular Chains </a></p>
-<p><a href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf" id="mc_basic_p6" target="_blank" rel="noopener noreferrer"> [8] Markov Chain:Basic Theory - Proposition 6 </a></p>
-<p><a href="https://arxiv.org/abs/1702.07339" id="fp_converse" target="_blank" rel="noopener noreferrer"> [9] A Converse to Banach's Fixed Point Theorem and its CLS Completeness </a></p>
-<p><a href="https://en.wikipedia.org/wiki/Cross-entropy#Cross-entropy_minimization" id="ce_kl" target="_blank" rel="noopener noreferrer"> [10] Cross-entropy minimization </a></p>
-<p><a href="https://thewolfsound.com/deconvolution-inverse-convolution/" id="deconv_1" target="_blank" rel="noopener noreferrer"> [11] Deconvolution Using Frequency-Domain Division </a></p>
-<p><a href="https://www.strollswithmydog.com/deconvolution-by-division-in-the-frequency-domain/" id="deconv_2" target="_blank" rel="noopener noreferrer"> [12] deconvolution-by-division-in-the-frequency-domain </a></p>
-<p><a href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf" id="mc_basic_t7" target="_blank" rel="noopener noreferrer"> [13] Markov Chain:Basic Theory - Theorem 7 </a></p>
-<p><a href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf" id="mc_basic_d4" target="_blank" rel="noopener noreferrer"> [14] Markov Chain:Basic Theory - Definition 4 </a></p>
-</span></div></div></div>

Markdown/zh/transform.html

@@ -1,53 +0,0 @@
-<div id="md_transform" class="block normal mds svelte-90oupt padded hide-container" style="border-style: solid; overflow: visible; min-width: min(0px, 100%); border-width: var(--block-border-width);"><div class="wrap center full svelte-1occ011 hide" style="position: absolute; padding: 0px;"></div> <div class="svelte-1ed2p3z"><div class="prose normal mds svelte-1yrv54" data-testid="markdown" dir="ltr"><span class="md svelte-1syupzx"><p>为了将初始的数据分布转换为简单的标准正态分布，扩散模型采用如下的变换方式
-<span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>=</mo><msqrt><mi>α</mi></msqrt><mi>X</mi><mo>+</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msqrt><mi>ϵ</mi><mspace width="2em"></mspace><mi>w</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>e</mi><mspace width="1em"></mspace><mi>α</mi><mo>&lt;</mo><mn>1</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>ϵ</mi><mo>∼</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}
-    Z = \sqrt{\alpha} X +  \sqrt{1-\alpha}\epsilon \qquad where \quad \alpha &lt; 1, \quad \epsilon \sim \mathcal{N}(0, I)  \tag{1.1}
-\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.5744em; vertical-align: -0.5372em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 1.0372em;"><span class="" style="top: -3.1228em;"><span class="pstrut" style="height: 3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8492em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.8092em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-其中<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X\sim q(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span>是任意的随机变量，<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo>∼</mo><mi>q</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z\sim q(Z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.0359em;">q</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right: 0.0715em;">Z</span><span class="mclose">)</span></span></span></span></span>是变换后的随机变量。</p>
-<p>此变换可分为两个子变换。</p>
-<p>第一个子变换是对随机变量<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span>执行一个线性变换(<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mi>α</mi></msqrt><mi>X</mi></mrow><annotation encoding="application/x-tex">\sqrt{\alpha}X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04em; vertical-align: -0.2397em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8003em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.7603em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.2397em;"><span class=""></span></span></span></span></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span>)，根据文献<a href="#linear_transform">[3]</a>的结论，线性变换使<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.0785em;">X</span></span></span></span></span>的概率分布“变窄变高”，并且"变窄变高"的程度与<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>的值成正比；具体可看Demo 1，左1图为随机生成的一维的概率分布，左2图是经过线性变换后的概率分布，可以看出，与左1图相比，左2图的曲线“变窄变高”了。读者可亲自测试不同的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>值，获得更直观的理解。</p>
-<p>第二个子变换是“加上独立的随机噪声”(<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msqrt><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\sqrt{1-\alpha}\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1.04em; vertical-align: -0.1744em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height: 0.8656em;"><span class="svg-align" style="top: -3em;"><span class="pstrut" style="height: 3em;"></span><span class="mord" style="padding-left: 0.833em;"><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span><span class="" style="top: -2.8256em;"><span class="pstrut" style="height: 3em;"></span><span class="hide-tail" style="min-width: 0.853em; height: 1.08em;"><svg width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
-c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
-c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
-c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
-s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
-c69,-144,104.5,-217.7,106.5,-221
-l0 -0
-c5.3,-9.3,12,-14,20,-14
-H400000v40H845.2724
-s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
-c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
-M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height: 0.1744em;"><span class=""></span></span></span></span></span><span class="mord mathnormal">ϵ</span></span></span></span></span>)，根据文献<a href="#sum_conv">[4]</a>的结论，“加上独立的随机变量”等效于对两个概率分布执行卷积，由于随机噪声的概率分布为高斯形状，所以相当于执行”高斯模糊“的操作。经过模糊后，原来的概率分布将变得更加平滑，与标准正态分布将更加相似。模糊的程度与噪声大小(<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.7278em; vertical-align: -0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>)正相关。具体可看Demo 1，左1图是随机生成的一维概率分布，左3图是经过变换后的结果，可以看出，变换后的曲线变光滑了，棱角变少了。读者可测试不同的<span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 0.4306em;"></span><span class="mord mathnormal" style="margin-right: 0.0037em;">α</span></span></span></span></span>值，感受噪声大小对概率分布曲线形状的影响。左4图是综合两个子变换后的结果。</p>
-</span></div></div></div>

Misc.py

@@ -1,14 +1,13 @@
  
 
-js_head = "<script>" + open("ExtraBlock.js").read() + "</script>"
-          # """ <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/katex@0.15.3/dist/katex.min.js" integrity="sha384-0fdwu/T/EQMsQlrHCCHoH10pkPLlKA1jL5dFyUOvB3lfeT2540/2g6YgSi2BL14p" crossorigin="anonymous"></script> """
+js_head = "<script>" + open("ExtraBlock.js").read() + "</script>" \
+          # + """ <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/katex@0.15.3/dist/katex.min.js" integrity="sha384-0fdwu/T/EQMsQlrHCCHoH10pkPLlKA1jL5dFyUOvB3lfeT2540/2g6YgSi2BL14p" crossorigin="anonymous"></script> """
 
 
 js_load = """
         function load_callback() {
             insert_markdown();
             add_switch();
-            insert_special_formula();
         }
           """
 

README.md

@@ -1,5 +1,5 @@
 ---
-title: DPM
+title: DPMInteractive
 emoji: 🌖
 colorFrom: red
 colorTo: gray

RenderMarkdownEn.py

@@ -1,6 +1,8 @@
 
 import gradio as gr
-from Misc import g_css, g_latex_del, js_head
+from Misc import g_css, js_head, g_latex_del
+js_head += """ <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/katex@0.15.3/dist/katex.min.js" integrity="sha384-0fdwu/T/EQMsQlrHCCHoH10pkPLlKA1jL5dFyUOvB3lfeT2540/2g6YgSi2BL14p" crossorigin="anonymous"></script> """
+
 
 
 def md_introduction_en():
@@ -10,17 +12,17 @@ def md_introduction_en():
         
         gr.Markdown(
             r"""
-            The Diffusion Model[\\[1\\]](#dpm)[\\[2\\]](#ddpm) is currently the main method used in image and video generation, but due to its abstruse theory, many engineers are unable to understand it well. This article will provide a very easy-to-understand method to help readers grasp the principles of the Diffusion Model. Specifically, it will illustrate the Diffusion Model using examples of one-dimensional random variables in an interactive way, explaining several interesting properties of the Diffusion Model in an intuitive manner.
+            The Diffusion Probability Model[\\[1\\]](#dpm)[\\[2\\]](#ddpm) is currently the main method used in image and video generation, but due to its abstruse theory, many engineers are unable to understand it well. This article will provide a very easy-to-understand method to help readers grasp the principles of the Diffusion Model. Specifically, it will illustrate the Diffusion Model using examples of one-dimensional random variables in an interactive way, explaining several interesting properties of the Diffusion Model in an intuitive manner.
             
             The diffusion model is a probabilistic model. Probabilistic models mainly offer two functions: calculating the probability of a given sample appearing; and generating new samples. The diffusion model focuses on the latter aspect, facilitating the production of new samples, thus realizing the task of **generation**.
             
-            The diffusion model differs from general probability models (such as GMM), which directly models the probability distribution of random variables. The diffusion model adopts an indirect approach, which utilizes **random variable transformation**(shown in Figure 1a) to gradually convert the data distribution (the probability distribution to be modeled) into the **standard normal distribution**, and meanwhile models the posterior probability distribution corresponding to each transformation (Figure 1b-c). Upon obtaining the final standard normal distribution and the posterior probability distributions, one can generate samples of each random variable $Z_T \ldots Z_2,Z_1,X$ in reverse order through Ancestral Sampling method. Simultaneously, initial data distribution $q(x)$ can be determined by employing Bayes theorem and the total probability theorem.
+            The diffusion model differs from general probability models (such as GMM), which directly models the probability distribution of random variables. The diffusion model adopts an indirect approach, which utilizes **random variable transform**(shown in Figure 1a) to gradually convert the data distribution (the probability distribution to be modeled) into the **standard normal distribution**, and meanwhile models the posterior probability distribution corresponding to each transformation (Figure 1b-c). Upon obtaining the final standard normal distribution and the posterior probability distributions, one can generate samples of each random variable $Z_T \ldots Z_2,Z_1,X$ in reverse order through <b>Ancestral Sampling</b>. Simultaneously, initial data distribution $q(x)$ can be determined by employing Bayes theorem and the total probability theorem.
             
             One might wonder: indirect methods require modeling and learning T posterior probability distributions, while direct methods only need to model one probability distribution, Why would we choose the indirect approach? Here's the reasoning: the initial data distribution might be quite complex and hard to represent directly with a probability model. In contrast, the complexity of each posterior probability distribution in indirect methods is significantly simpler, allowing it to be approximated by simple probability models. As we will see later, given certain conditions, posterior probability distributions can closely resemble Gaussian distributions, thus a simple conditional Gaussian model can be used for modeling.
             
-            <center> <img src="file/pipe.jpg" width="820" style="margin-top:12px"/> </center>
-            <center> Figure 1: Diffusion model schematic </center>
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_introduction")
+            <center> <img id="en_fig1" src="file/fig1.png" width="820" style="margin-top:12px"/> </center>
+            <center> Figure 1: Diffusion probability model schematic </center>
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_introduction_en")
     return
 
 
@@ -46,7 +48,7 @@ def md_transform_en():
             The second sub-transformation is **adding independent random noise**($\sqrt{1-\alpha}\epsilon$). According to the conclusion of the literature[\[4\]](#sum_conv), **adding independent random variables** is equivalent to performing convolution on the two probability distributions. Since the probability distribution of random noise is Gaussian, it is equivalent to performing a **Gaussian Blur** operation. After blurring, the original probability distribution will become smoother and more similar to the standard normal distribution. The degree of blurring is directly proportional to the noise level ($\sqrt{1-\alpha}$).
             
             For specifics, one can see Demo 1, where the first figure is a randomly generated one-dimensional probability distribution, and the third  figure is the result after the transformation. It can be seen that the transformed probability distribution curve is smoother and there are fewer corners. The readers can test different $\alpha$ values to feel how the noise level affect the shape of the probability distribution. The last figure is the result after applying all two sub-transformations.
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_transform")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_transform_en")
     return
 
 
@@ -62,7 +64,7 @@ def md_likelihood_en():
                 q(z|x) &= \mathcal{N}(\sqrt{\alpha}x,\ 1-\alpha)    \tag{2.1}
             \end{align}
             It can be understood by concrete examples in Demo 2. The third figure depict the shape of $q(z|x)$. From the figure, a uniform slanting line can be observed. This implies that the mean of $q(z|x)$ is linearly related to x, and the variance is fixed. The magnitude of $\alpha$ will determine the width and incline of the slanting line.
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_likelihood")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_likelihood_en")
     return
 
 
@@ -105,11 +107,11 @@ def md_posterior_en():
             
             The posterior $q(x|z)$ under two special states are worth considering.
             <ul>
-            <li>As $\alpha \to 0$, the variance of <b>GaussFun</b> tends to <b>$\infty$</b>, and $q(x|z)$ for different $z$ almost become identical, and almost the same as $q(x)$. Readers can set $\alpha$ to 0.01 in Demo 2 to observe the specific results.</li>
+            <li>As $\alpha \to 0$, the variance of <b>GaussFun</b> tends to <b>$\infty$</b>, and $q(x|z)$ for different $z$ almost become identical, and almost the same as $q(x)$. Readers can set $\alpha$ to 0.001 in Demo 2 to observe the specific results.</li>
                 
             <li>As $\alpha \to 1$, the variance of <b>GaussFun</b> tends to <b>$0$</b>, The $q(x|z)$ for different $z$ values contract into a series of <em>Dirac delta functions</em> with different offsets equalling to $z$. However, there are some exceptions. When there are regions where $q(x)$ is zero, the corresponding $q(x|z)$ will no longer be a Dirac <em>delta function</em>, but a zero function. Readers can set $\alpha$ to 0.999 in Demo 2 to observe the specific results.</li>
             </ul>
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_en")
     return
 
 
@@ -163,7 +165,7 @@ def md_forward_process_en():
                 Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon 			 \tag{4.10}
             \end{align}
             It can be seen that, after applying two transforms, the transformed distributions $q(z_T|x)$ are the same. Thus, $q(z_T)$ is also the same. 
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_forward_process")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_forward_process_en")
     return
 
 
@@ -202,7 +204,7 @@ def md_backward_process_en():
             Since $\\alpha=0.0000403$ is very small, the corresponding standard deviation of GaussFun (Equation 3.4) reaches 157.52. However, the range of $X$ is limited within $[-1, 1]$, which is far smaller than the standard deviation of GaussFun. Within the range of $x \\in [-1, 1]$, GaussFun should be close to a constant, showing little variation. Therefore, the $q(x|z_T)$ corresponding to different $z_T$ are almost the same as $q(x)$. In this state, the posterior transform corresponding to $q(x|z_T)$ does not depend on the input distribution, the output distribution will always be $q(x)$.
             
             <b>Therefore, theoretically, in the DDPM model, it is not necessary to use the standard normal distribution to replace $q(z_T)$. Any other arbitrary distributions can also be used as a substitute.</b>
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_backward_process")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_backward_process_en")
     return
 
 
@@ -220,9 +222,9 @@ def md_fit_posterior_en():
             
             Due to the limitations of the model's representative and learning capabilities, there will be certain errors in the fitting process, which will further impact the accuracy of restored $q(x)$. The size of the fitting error is related to the complexity of the posterior probability distribution. As can be seen from Section 4, when $q(x)$ is more complex or the added noise is large, the posterior probability distribution will be more complex, and it will differ greatly from the Gaussian distribution, thus leading to fitting errors and further affecting the restoration of $q(x)$.
             
-            Refer to Demo 3.3 for the specifics. The reader can test different $q(x)$ and $\alpha$, observe the fitting degree of the posterior probability distribution $q(z_{t-1}|z_t)$ and the accuracy of restored $q(x)$. The restored probability distribution is marked with $\textcolor{orange}{\text{orange}}$, and the error is also indicated by JS divergence.
+            Refer to Demo 3.3 for the specifics. The reader can test different $q(x)$ and $\alpha$, observe the fitting degree of the posterior probability distribution $q(z_{t-1}|z_t)$ and the accuracy of restored $q(x)$. The restored probability distribution is ploted with $\textcolor{orange}{\text{orange}}$, and the error is also measured by JS divergence.
              
-            Regarding the objective function for fitting, similar to other probability models, the cross-entropy loss can be optimized to make $p(z_{t-1}|z_t)$ approaching $q(z_{t-1}|z_t)$. Since $(z_{t-1}|z_t)$ is a conditional probability, it is necessary to fully consider all conditions. This can be achieved By Averaging the cross-entropy corresponding to each condition weighted by the probability of each condition happening. The final form of the loss function is as follows.
+            Regarding the objective function for fitting, similar to other probability models, the cross-entropy loss can be optimized to make $p(z_{t-1}|z_t)$ approaching $q(z_{t-1}|z_t)$. Since $(z_{t-1}|z_t)$ is a conditional probability, it is necessary to fully consider all conditions. This can be achieved by averaging the cross-entropy corresponding to each condition weighted by the probability of each condition happening. The final form of the loss function is as follows.
             \begin{align}
                 loss &= -\int q(z_t) \overbrace{\int q(z_{t-1}|z_t) \log \textcolor{blue}{p(z_{t-1}|z_t)}dz_{t-1}}^{\text{Cross Entropy}}\ dz_t     \tag{6.1}     \newline
                      &= -\iint q(z_{t-1},z_t) \log \textcolor{blue}{p(z_{t-1}|z_t)}dz_{t-1}dz_t                                                     \tag{6.2} 
@@ -241,7 +243,7 @@ def md_fit_posterior_en():
                      &\approx -\sum_{i=0}^N \log \textcolor{blue}{p(Z_{t-1}^i|Z_t^i)} \qquad where \quad (Z_{t-1}^i,Z_t^i) \sim q(z_{t-1},z_t)      \tag{6.7} 
             \end{align}
 
-            The aforementioned samples $(Z_{t-1}^i,Z_t^i)$ follow a joint probability distribution $q(z_{t-1},z_t)$, which can be sampled via an <b>Ancestral Sampling</b>. The specific method is as follows: sample $X,Z_1,Z_2 \dots Z_{t-1},Z_t$ step by step through forward transformation (Formulas 4.1~4.4), and then reserve $(Z_{t-1},Z_t)$ as a sample. This sampling process is relatively slow. To speed up the sampling, we can take advantage of the known features of the probability distribution $q(z_t|x)$ (Formula 4.8). First, sample $X$ from $q(x)$, then sample $Z_{t-1}$ from $q(z_{t-1}|x)$, and finally sample $Z_t$ from $q(z_t|z_{t-1})$. Thus, a sample $(Z_{t-1},Z_t)$ is obtained.
+            The aforementioned samples $(Z_{t-1}^i,Z_t^i)$ follow a joint probability distribution $q(z_{t-1},z_t)$, which can be sampled via an <b>Ancestral Sampling</b>. The specific method is as follows: sample $X,Z_1,Z_2 \dots Z_{t-1},Z_t$ step by step through forward transforms (Formulas 4.1~4.4), and then reserve $(Z_{t-1},Z_t)$ as a sample. This sampling process is relatively slow. To speed up the sampling, we can take advantage of the known features of the probability distribution $q(z_t|x)$ (Formula 4.8). First, sample $X$ from $q(x)$, then sample $Z_{t-1}$ from $q(z_{t-1}|x)$, and finally sample $Z_t$ from $q(z_t|z_{t-1})$. Thus, a sample $(Z_{t-1},Z_t)$ is obtained.
             
             Some people may question that the objective function in Equation 6.3 seems different from those in the DPM[\\[1\\]](#dpm) and DDPM[\\[2\\]](#ddpm) papers. In fact, these two objective functions are equivalent, and the proof is given below. 
 
@@ -281,7 +283,7 @@ def md_fit_posterior_en():
                 \mathop{\min}{underline}{\textcolor{blue}{p}}  KL(q(z)\|\textcolor{blue}{p(z)})  \iff  \mathop{\min}_{\textcolor{blue}{p}} \int \ q(x) KL(q(z|x)\|\textcolor{blue}{p(z)})dx         \tag{6.20}
             </span>
             
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_fit_posterior")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_fit_posterior_en")
     return
 
 
@@ -321,7 +323,7 @@ def md_posterior_transform_en():
             \end{align}
             In order to better understand the property of the transform, the matrix $(Q_{x|z})^n$ is also plotted in Demo 4.2. From the demo we can see that, as the iterations converge, the row vectors of the matrix $(Q_{x|z})^n$ will become a constant vector, that is, all components of the vector will be the same, which will appear as a horizontal line in the denisty plot.
             
-            <center> <img src="file/fig2.png" width="960" style="margin-top:12px"/> </center>
+            <center> <img id="en_fig2" src="file/fig2.png" width="960" style="margin-top:12px"/> </center>
             <center> Figure 2: Only one component in support </center>
             
             The following will prove that with some conditions, the posterior transform is a contraction mapping, and there exists a unique point, which is also the converged point. The proof assumes that the random variable is discrete, so the posterior transform can be regarded as a single step transition of a <b>discrete Markov Chain</b>. The posterior $q(x|z)$ corresponds to the <b>transfer matrix</b>. Continuous variables can be considered as discrete variables with infinite states.
@@ -331,13 +333,13 @@ def md_posterior_transform_en():
             <li> When $q(x)$ is partially greater than 0, and the support of $q(x)$ (the region where $q(x)$ is greater than 0) consists only one connected component (Figure 2), several conclusions can be drawn from equation (3.4)：
             
             <ol style="list-style-type:lower-alpha; padding-inline-start: 0px;font-size:16px;">
-            <li> When $z$ and $x$ are within the support set, since both $q(x)$ and GaussFun are greater than 0, the diagonal elements of the transfer matrix $\\{q(x|z)|z=x\\}$ are greater than 0. This means that the state within the support set is $\textcolor{green}{\text{aperiodic}}$. </li>
+            <li> When $z$ and $x$ are within the support set, since both $q(x)$ and GaussFun are greater than 0, the diagonal elements of the transfer matrix $\{q(x|z)|z=x\}$ are greater than 0. This means that the state within the support set is $\textcolor{green}{\text{aperiodic}}$. </li>
             
             <li> When $z$ and $x$ are within the support set, since GaussFun's support set has a certain range, elements above and below the diagonal $\{q(x|z)|x=z+\epsilon\}$is also greater than 0. This means that states within the support set are accessible to each other, forming a $\textcolor{red}{\text{Communication Class}}$<a href="#mc_basic_d4">[14]</a>, see in Figure 2b. </li>
                 
             <li> When <em>$z$ is within the support set</em> and <em>$x$ is outside the support set</em>, ${q(x|z)}$ is entirely 0. This means that the state within the support set is <em>inaccessible</em> to the state outside the support set (Inaccessible Region in Figure 2b) </li>
                 
-            <li> When <em>$z$ is outside the support set</em> and <em>$x$ is inside the support set</em>, due to the existence of a certain range of the support set of GaussFun, there are some extension areas (Extension Region in Figure 2b), where the corresponding $\{q(x|z)|x \in support\}$ is not all zero. This means that the state of this part of the extension area can <em>unidirectionally</em> access (access) the state inside the support set (Unidirectional Region in Figure 2b).</li>
+            <li> When <em>$z$ is outside the support set</em> and <em>$x$ is inside the support set</em>, due to the existence of a certain range of the support set of GaussFun, there are some extension areas (Extension Region in Figure 2b), where the corresponding $\{q(x|z)|x \in support\}$ is not all zero. This means that the state of this part of the extension area can <em>unidirectionally</em> access the state inside the support set (Unidirectional Region in Figure 2b).</li>
                 
             <li> When <em>$z$ is outside the support set</em> and <em>$x$ is outside the support set</em>, the corresponding $q(x|z)$ is entirely zero. This implies that, states outside the support set will not transit to states outside the support set. In other words, states outside the support set only originate from states within the support set. </li>
             
@@ -356,10 +358,10 @@ def md_posterior_transform_en():
             <li> When $q(x)$ is partially greater than 0, and multiple connected component exist in the support set of $q(x)$, and the maximum distance of each connected component <b>cannot</b> be covered by the support set of corresponding GaussFun, the states within each component <b>constitute multiple Communicate Classes</b>, as shown in Figure 4. Under such circumstances, as $n\to\infty$, $q(x|z)^n$ will also converge to a fixed matrix, but not all the column vectors are identical. Therefore, the posterior transforma is not a strict contraction mapping. However, when the state of the input distribution is confined to a single Communicate Class and its corresponding extension, the posterior transform is also a contraction mapping with a unique convergence point. </li>
             </ol>
             
-            <center> <img src="file/fig3.png" width="960" style="margin-top:12px"/> </center>
+            <center> <img id="en_fig3" src="file/fig3.png" width="960" style="margin-top:12px"/> </center>
             <center> Figure 3: Two components which can communicate with each other </center>
             
-            <center> <img src="file/fig4.png" width="960" style="margin-top:12px"/> </center>
+            <center> <img id="en_fig4" src="file/fig4.png" width="960" style="margin-top:12px"/> </center>
             <center> Figure 4: Two components which <b>cannot</b> communicate with each other </center>
             
             Additionally, there exists a more generalized relation about the posterior transform that is independent of $q(x|z)$: the Total Variance distance between two output distributions will always be <b>less than or equal to</b> the Total Variance distance between their corresponding input distributions, that is
@@ -392,7 +394,7 @@ def md_posterior_transform_en():
             
             From the above discussion, we know that the smaller the $\alpha$ (the larger the noise used in the transform process), the greater the contractive ratio of the contraction mapping, and thus, the stronger the ability to resist noise.
             
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform_en")
     return
 
 
@@ -408,7 +410,7 @@ def md_deconvolution_en():
             However, in actuality, some problems do exist. Due to the extreme sensitivity of deconvolution to errors, having high input sensitivity, even a small amount of input noise can lead to significant changes in output[\[11\]](#deconv_1)[\[12\]](#deconv_2). Meanwhile, in the diffusion model, the standard normal distribution is used as an approximation to replace $q(z_T)$, thus, noise is introduced at the initial stage of recovery. Although the noise is relatively small, because of the sensitivity of deconvolution, the noise will gradually amplify, affecting the recovery.
             
             In addition, the infeasibility of <b>deconvolution restoring</b> can be understood from another perspective. Since the process of forward transform (equations 4.1 to 4.4) is fixed, the convolution kernel is fixed. Therefore, the corresponding deconvolution transform is also fixed. Since the initial data distribution $q(x)$ is arbitrary, any probability distribution can be transformed into an approximation of $\mathcal{N}(0,I)$ through a series of fixed linear transforms and convolutions. If <b>deconvolution restoring</b> is feasible, it means that a fixed deconvolution can be used to restore any data distribution $q(x)$ from the $\mathcal{N}(0,I)$ , this is clearly <b>paradoxical</b>. The same input, the same transform, cannot have multiple different outputs.
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution_en")
     return
 
 
@@ -447,7 +449,7 @@ def md_reference_en():
             
             <a id="mc_basic_d4" href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf"> [14] Markov Chain:Basic Theory - Definition 4 </a>
              
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_reference")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_reference_en")
 
     return
 
@@ -464,7 +466,7 @@ def md_about_en():
             <b>Author</b>: Zhenxin Zheng, Senior computer vision engineer with ten years of algorithm development experience, Formerly employed by Tencent and JD.com, currently focusing on image and video generation.
             
             <b>Email</b>: blair.star@163.com.
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_about")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_about_en")
 
     return
 

RenderMarkdownZh.py

@@ -1,6 +1,8 @@
 
 import gradio as gr
-from Misc import g_css, g_latex_del, js_head
+from Misc import g_css, js_head, g_latex_del
+js_head += """ <script type="text/javascript" src="https://cdn.jsdelivr.net/npm/katex@0.15.3/dist/katex.min.js" integrity="sha384-0fdwu/T/EQMsQlrHCCHoH10pkPLlKA1jL5dFyUOvB3lfeT2540/2g6YgSi2BL14p" crossorigin="anonymous"></script> """
+
 
 
 def md_introduction_zh():
@@ -14,13 +16,13 @@ def md_introduction_zh():
 
             扩散模型是一个概率模型。概率模型主要提供两方面的功能：计算给定样本出现的概率；采样生成新样本。扩散模型侧重于第二方面，方便采样新样本，从而实现"生成"的任务。
 
-            扩散模型与一般的概率模型(如GMM)不同，直接建模随机变量的概率分布。扩散模型采用一种间接方式，利用“随机变量变换”的方式(如图1a)，逐步将待建模的概率分布(数据分布)转变成"标准正态分布"，同时，建模学习各个变换对应的后验概率分布(图1b-c)。有了最终的标准正态分布和各个后验概率分布，则可通过祖先采样的方式，从反向逐步采样得到各个随机变量$Z_T \ldots Z_2,Z_1,X$的样本。同时也可通过贝叶斯公式和全概率公式确定初始的数据分布$q(x)$。
+            扩散模型与一般的概率模型(如GMM)不同，直接建模随机变量的概率分布。扩散模型采用一种间接方式，利用“随机变量变换”的方式(如图1a)，逐步将待建模的概率分布(数据分布)转变成"标准正态分布"，同时，建模学习各个变换对应的后验概率分布(图1b-c)。有了最终的标准正态分布和各个后验概率分布，则可通过祖先采样(Ancestral Sampling)的方式，从反向逐步采样得到各个随机变量$Z_T \ldots Z_2,Z_1,X$的样本。同时也可通过贝叶斯公式和全概率公式确定初始的数据分布$q(x)$。
 
             可能会有这样的疑问：间接的方式需要建模学习T个后验概率分布，直接方式只需要建模学习一个概率分布，为什么要选择间接的方式呢？是这样子的：初始的数据分布可能很复杂，很难用一个概率模型直接表示；而对于间接的方式，各个后验概率分布的复杂度会简单许多，可以用简单的概率模型进行拟合。下面将会看到，当满足一些条件时，后验概率分布将非常接近高斯分布，所以可以使用简单的条件高斯模型进行建模。
 
-            <center> <img src="file/pipe.jpg" width="820" style="margin-top:12px"/> </center>
+            <center> <img src="file/fig1.png" width="820" style="margin-top:12px"/> </center>
             <center> Figure 1: Diffusion model schematic </center>
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_introduction")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_introduction_zh")
     return
 
 
@@ -42,7 +44,7 @@ def md_transform_zh():
             第一个子变换是对随机变量$X$执行一个线性变换($\sqrt{\alpha}X$)，根据文献[\[3\]](#linear_transform)的结论，线性变换使$X$的概率分布“变窄变高”，并且"变窄变高"的程度与$\alpha$的值成正比；具体可看Demo 1，左1图为随机生成的一维的概率分布，左2图是经过线性变换后的概率分布，可以看出，与左1图相比，左2图的曲线“变窄变高”了。读者可亲自测试不同的$\alpha$值，获得更直观的理解。
 
             第二个子变换是“加上独立的随机噪声”($\sqrt{1-\alpha}\epsilon$)，根据文献[\[4\]](#sum_conv)的结论，“加上独立的随机变量”等效于对两个概率分布执行卷积，由于随机噪声的概率分布为高斯形状，所以相当于执行”高斯模糊“的操作。经过模糊后，原来的概率分布将变得更加平滑，与标准正态分布将更加相似。模糊的程度与噪声大小($1-\alpha$)正相关。具体可看Demo 1，左1图是随机生成的一维概率分布，左3图是经过变换后的结果，可以看出，变换后的曲线变光滑了，棱角变少了。读者可测试不同的$\alpha$值，感受噪声大小对概率分布曲线形状的影响。左4图是综合两个子变换后的结果。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_transform")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_transform_zh")
     return
 
 
@@ -58,7 +60,7 @@ def md_likelihood_zh():
                 q(z|x) &= \mathcal{N}(\sqrt{\alpha}x,\ 1-\alpha)    \tag{2.1}
             \end{align}
             具体可看Demo 2，左3图展示了$q(z|x)$的形状，从图中可以看到一条均匀的斜线，这意味着$q(z|x)$的均值与x线性相关，方差固定不变。$\alpha$值的大小将决定斜线宽度和倾斜程度。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_likelihood")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_likelihood_zh")
     return
 
 
@@ -101,7 +103,7 @@ def md_posterior_zh():
                 <li> 当$\alpha \to 0$时，GaussFun的方差趋向于<b>无穷大</b>，不同$z$值的$q(x|z)$几乎变成一致，并与$q(x)$几乎相同。读者可在Demo 2中，将$\alpha$设置为0.01，观察具体的结果。</li>
                 <li> 当$\alpha \to 1$时，GaussFun的方差趋向于<b>无穷小</b>，不同$z$值的$q(x|z)$收缩成一系列不同偏移量的Dirac delta函数, 偏移量等于$z$。但有一些例外，当q(x)存在为零的区域时，其对应的q(x|z)将不再为Dirac delta函数，而是零函数。可在Demo 2中，将$\alpha$设置为0.999，观察具体的结果。</li>
             </ul>
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_zh")
     return
 
 
@@ -155,7 +157,7 @@ def md_forward_process_zh():
                 Z_T = \sqrt{0.0000403}\ X + \sqrt{1-0.0000403}\ \epsilon = 0.00635\ X + 0.99998\ \epsilon 			 \tag{4.10}
             \end{align}
             可以看出，应用两种变换后，变换后的分布$q(z_T|x)$相同，因此，$q(z_T)$也相同。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_forward_process")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_forward_process_zh")
     return
 
 
@@ -194,7 +196,7 @@ def md_backward_process_zh():
             
             <b>所以，理论上，在DDPM模型中，无需非得使用标准正态分布代替$q(z_T)$，也可使用其它任意的分布代替。</b>
              
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_backward_process")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_backward_process_zh")
     return
 
 
@@ -271,7 +273,7 @@ def md_fit_posterior_zh():
             <span id="zh_fit_2">
                 \mathop{\min}{underline}{\textcolor{blue}{p}}  KL(q(z)\|\textcolor{blue}{p(z)})  \iff  \mathop{\min}_{\textcolor{blue}{p}} \int \ q(x) KL(q(z|x)\|\textcolor{blue}{p(z)})dx
             </span>
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_fit_posterior")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_fit_posterior_zh")
     return
 
 
@@ -379,7 +381,7 @@ def md_posterior_transform_zh():
             具体可看Demo 3.2，通过增加“noise ratio”的值可以向“末尾分布$q(z_T)$”添加噪声，点击“apply”按钮将逐步画出恢复的过程，恢复的分布以$\textcolor{red}{红色曲线}$画出，同时也会通过JS散度标出误差的大小。将会看到，恢复的$q(x)$的误差总是小于$q(z_T)$的误差。
             
             由上面的讨论可知，$\alpha$越小(即变换过程中使用的噪声越大)，压缩映射的压缩率越大，于是，抗噪声的能力也越强。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_posterior_transform_zh")
     return
 
 
@@ -396,7 +398,7 @@ def md_deconvolution_zh():
             但实际上，会存在一些问题。由于逆卷积对误差极为敏感，具有很高的输入灵敏度，很小的输入噪声就会引起输出极大的变化[\[11\]](#deconv_1)[\[12\]](#deconv_2)。而在扩散模型中，会使用标准正态分布近似代替$q(z_T)$，因此，在恢复的起始阶段就会引入噪声。虽然噪声较小，但由于逆卷积的敏感性，噪声会逐步放大，影响恢复。
             
             另外，也可以从另一个角度理解“逆卷积恢复”的不可行性。由于前向变换的过程(式4.1~4.4)是确定的，所以卷积核是固定的，因此，相应的“逆卷积变换“也是固定的。由于起始的数据分布$q(x)$可以是任意的分布，所以，通过一系列固定的“卷积正变换”，可以将任意的概率分布转换成近似$\mathcal{N}(0,I)$的分布。如“逆卷积变换“可行，则意味着，可用一个固定的“逆卷积变换"，将$\mathcal{N}(0,I)$分布恢复成任意的数据分布$q(x)$，这明显是一个悖论。同一个输入，同一个变换，不可能会有多个输出。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_deconvolution_zh")
     return
 
 
@@ -436,7 +438,7 @@ def md_reference_zh():
             
             <a id="mc_basic_d4" href="http://galton.uchicago.edu/~lalley/Courses/312/MarkovChains.pdf"> [14] Markov Chain:Basic Theory - Definition 4 </a>
             
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_reference")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_reference_zh")
 
     return
 
@@ -453,14 +455,14 @@ def md_about_zh():
             <b>Author</b>: 郑镇鑫，资深视觉算法工程师，十年算法开发经历，曾就职于腾讯京东等互联网公司，目前专注于视频生成(类似Sora)。
             
             <b>Email</b>: blair.star@163.com 。
-            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_about")
+            """, latex_delimiters=g_latex_del, elem_classes="normal mds", elem_id="md_about_zh")
 
     return
 
 
 def run_app():
     
-    # with gr.Blocks(css=g_css, js="() => insert_special_formula()", head=js_head) as demo:
+    # with gr.Blocks(css=g_css, js="() => insert_special_formula(); ", head=js_head) as demo:
     with gr.Blocks(css=g_css, js="() => {insert_special_formula(); write_markdown();}", head=js_head) as demo:
         md_introduction_zh()