Upload Introduction.md

Introduction.md

@@ -9,15 +9,37 @@ Conformal prediction is a technique for quantifying such uncertainties for AI sy
 
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-#### Theory
+# Theory
 
 ### 1. Prediction Regions
 
-Prediction regions represent intervals that contain the true value of the prediction with a certain confidence level. In regression, this is often expressed as a prediction interval. Let's denote the prediction region as \[$a$, $b$\], where $a$ and $b$ are the lower and upper bounds, respectively. The confidence level is denoted by $\alpha$. In classification, the prediction region is a set of classes that's above a certain threshold. The threshold is calculated by $\alpha$.
+Prediction regions in conformal prediction are intervals that provide a range of possible values for the prediction. For a regression task, this is often referred to as a prediction interval. Let's denote the prediction region as $[a, b]$, where $a$ and $b$ represent the lower and upper bounds, respectively. The confidence level is denoted by $\alpha$. The prediction region is constructed in such a way that it contains the true value with a probability of at least $(1 - \alpha)$.
+
+Mathematically, for a prediction $\hat{y}$, the prediction region is defined as:
+
+$$ P(a \leq y \leq b) \geq 1 - \alpha $$
+
+This ensures that the true value $y$ falls within the predicted interval with a confidence level of at least $(1 - \alpha)$.
+
+For a classification task, the prediction region is a set of classes that's above a certain threshold. The threshold is calculated by $\alpha$. Mathematically, for a prediction $\hat{C}$, the prediction region is defined as:
+
+$$ P(y \in \hat{C}) \geq 1 - \alpha $$
+
+This ensures that the true value $y$ falls within the predicted set of classes with a confidence level of at least $(1 - \alpha)$.
 
 ### 2. Validity
 
-A conformal predictor is considered valid if the true value falls within the predicted region with the specified confidence level over repeated experiments. Mathematically, for a given prediction $\hat{y}$ and a true outcome $y$, the validity condition is given by $P(y \in [a, b]) \geq 1 - \alpha$. In classification, the validity condition is given by $P(y \in \hat{C}) \geq 1 - \alpha$, where $\hat{C}$ is the predicted set of classes.
+The validity of a conformal predictor is a crucial aspect. It ensures that, over repeated experiments, the true value falls within the predicted region with the specified confidence level. Mathematically, for a given prediction $\hat{y}$ and a true outcome $y$, the validity condition is expressed as:
+
+$$ P(y \in [a, b]) \geq 1 - \alpha $$
+
+This means that the probability of the true value $y$ lying within the predicted interval $[a, b]$ is greater than or equal to $(1 - \alpha)$.
+
+For a classification task, the validity condition is expressed as:
+
+$$ P(y \in \hat{C}) \geq 1 - \alpha $$
+
+This means that the probability of the true value $y$ lying within the predicted set of classes $\hat{C}$ is greater than or equal to $(1 - \alpha)$.
 
 ### 3. Inductive Conformal Prediction
 
@@ -36,5 +58,3 @@ Inductive Conformal Prediction is characterized by its adaptability to the data
     - validity score $s_i$ by counting the number of times the true value $y_i$ falls within the prediction region $R_i$ over repeated experiments.
     - p-value $p_i$ by dividing the validity score $s_i$ by the number of repeated experiments.
     - prediction region $R_i$ using the top $k$ errors and the p-value $p_i$.
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