bayesian-statistics-ai999x: offline, verified bayesian statistics solver with step-by-step answers

Desktop app & command line · Compute in your browser · Install (Python) · Guides · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.5.0

Field: Mathematics & Statistics · Tasks: Question Answering

bayesian-statistics-ai999x is an offline, deterministic bayesian statistics solver written in Rust: a desktop app, a command-line program, a browser module and a Python package. It answers bayesian statistics questions with verified results from 78 certified capabilities, gives 378 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

bayesian-statistics-ai999x reads a question in plain English (or a typed call), picks a certified bayesian statistics capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions, or a refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Run it your way. bayesian-statistics-ai999x comes as the AI999X desktop app and as the command-line program bayesian-statistics for Windows and Linux, runs in your browser as WebAssembly, and installs as a Python package. The desktop app, the command line and the browser run the same compiled Rust program on your own device: offline, no Python needed.

Get it

Way to run it	For	Get it
AI999X Bayesian Statistics desktop app 1.5.0	Windows, Linux. This model in its own app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
AI999X desktop app 1.5.0	Windows, Linux. Every AI999X model in one app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
Command line (bayesian-statistics)	Windows, Linux. One program per model, offline, no Python	Windows (x86-64) · Linux (x86-64)
In your browser	any device. Nothing to install	Compute on the model page
Python	Python 3.10+	pip install bayesian-statistics-ai999x

Checksums: SHA256SUMS · every version: https://downloads.ai999x.com/desktop/latest/ · install guide: desktop & command line

Quick start

Desktop app

Download AI999X Bayesian Statistics for your system (above), unpack it and start it; type a question in words or pick a capability, fill in the form and press Compute. Figures and an explanation of the answer appear beside it. The AI999X desktop app has the same, with Bayesian Statistics among all AI999X models.

Linux, from a terminal:

curl -LO https://downloads.ai999x.com/desktop/1.5.0/ai999x-bayesian-statistics-1.5.0-linux-x86_64.tar.gz
curl -LO https://downloads.ai999x.com/desktop/1.5.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
tar xzf ai999x-bayesian-statistics-1.5.0-linux-x86_64.tar.gz
./ai999x-bayesian-statistics-1.5.0-linux-x86_64/ai999x-bayesian-statistics

Command line

Linux:

curl -LO https://downloads.ai999x.com/desktop/1.5.0/ai999x-cli-1.5.0-linux-x86_64.tar.gz
curl -LO https://downloads.ai999x.com/desktop/1.5.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
mkdir -p ~/.local/share && tar xzf ai999x-cli-1.5.0-linux-x86_64.tar.gz -C ~/.local/share
echo 'export PATH="$HOME/.local/share/ai999x-cli-1.5.0-linux-x86_64:$PATH"' >> ~/.bashrc && source ~/.bashrc
bayesian-statistics --version

Windows (PowerShell):

$F = "ai999x-cli-1.5.0-windows-x86_64.zip"
Invoke-WebRequest "https://downloads.ai999x.com/desktop/1.5.0/$F" -OutFile $F
Invoke-WebRequest "https://downloads.ai999x.com/desktop/1.5.0/SHA256SUMS" -OutFile SHA256SUMS
(Get-FileHash $F -Algorithm SHA256).Hash.ToLower() -eq ((Select-String -SimpleMatch $F SHA256SUMS).Line -split ' ')[0]
Expand-Archive $F -DestinationPath "$env:LOCALAPPDATA\AI999X" -Force
$D = "$env:LOCALAPPDATA\AI999X\ai999x-cli-1.5.0-windows-x86_64"; $env:Path += ";$D"
[Environment]::SetEnvironmentVariable("Path", [Environment]::GetEnvironmentVariable("Path", "User") + ";$D", "User")
bayesian-statistics --version

Every command below was run as shown; the text under it is what the program printed.

Ask in words: Write the question as you would say it: the program reads the quantities and their units, picks the capability and answers.

$ bayesian-statistics ask "A test has sensitivity 0.9 and specificity 0.95; the prevalence is 0.01. What is the positive predictive value?"
Read as: Positive and negative predictive values of a test
ppv = 0.1538461538
npv = 0.9989378651
p_positive = 0.0585
false_discovery_rate = 0.8461538462
false_omission_rate = 0.001062134891
  by: Positive and negative predictive values of a test  (Bayes' theorem with the total probability of a positive result)
  prevalence = 0.01
  sensitivity = 0.9
  specificity = 0.95
  assumes: probabilities between 0 and 1; both results possible

What a capability needs: The equation or method, every input and output with its SI unit. Read this before solve: it tells you the input names.

$ bayesian-statistics info diagnostic_test_predictive_values
Positive and negative predictive values of a test
  Bayes' theorem with the total probability of a positive result
  in   prevalence prevalence (pre-test probability of the condition) [1]
  in   sensitivity sensitivity P(positive | condition) [1]
  in   specificity specificity P(negative | no condition) [1]
  out  ppv      positive predictive value P(condition | positive) [1]
  out  npv      negative predictive value P(no condition | negative) [1]
  out  p_positive probability of a positive result [1]
  out  false_discovery_rate 1 - PPV [1]
  out  false_omission_rate 1 - NPV [1]

Solve: Inputs as name=value, each with any unit of the right kind. The answer lists every output, the method used, each input converted to SI, and the conditions the answer assumes.

$ bayesian-statistics solve diagnostic_test_predictive_values prevalence=0.01 sensitivity=0.9 specificity=0.95
ppv = 0.1538461538
npv = 0.9989378651
p_positive = 0.0585
false_discovery_rate = 0.8461538462
false_omission_rate = 0.001062134891
  by: Positive and negative predictive values of a test  (Bayes' theorem with the total probability of a positive result)
  prevalence = 0.01
  sensitivity = 0.9
  specificity = 0.95
  assumes: probabilities between 0 and 1; both results possible

Many questions at once: One JSON request per line in, one JSON answer per line out, in order: for scripts, notebooks and agents (the program starts once). In PowerShell: Get-Content cases.jsonl | bayesian-statistics batch

cases.jsonl:

{"key": "diagnostic_test_predictive_values", "inputs": {"prevalence": 0.01, "sensitivity": 0.9, "specificity": 0.95}}
{"key": "beta_binomial_posterior", "inputs": {"alpha": 2.0, "beta": 2.0, "successes": "7", "trials": "10"}}

$ bayesian-statistics batch < cases.jsonl
{"outcome":"SOLVED","message":"","value":0.153846153846,"unit":"","solved_for":"ppv","si":[["prevalence",0.01],["sensitivity",0.9],["specificity",0.95]],"values":[["ppv",0.153846153846,""],["npv",0.998937865109,""],["p_positive",0.0585,""],["false_discovery_rate",0.846153846154,""],["false_omission_rate",0.00106213489113,""]],"target":"ppv","phrase":"positive predictive value P(condition | positive)","assumptions":["probabilities between 0 and 1","both results possible"],"name":"Positive and negative predictive values of a test","equation":"Bayes' theorem with the total probability of a positive result"}
{"outcome":"SOLVED","message":"","value":0.642857142857,"unit":"","solved_for":"posterior_mean","si":[["alpha",2.0],["beta",2.0]],"values":[["post_alpha",9.0,""],["post_beta",5.0,""],["posterior_mean",0.642857142857,""],["posterior_variance",0.015306122449,""],["posterior_sd",0.123717914826,""]],"target":"posterior_mean","phrase":"posterior alpha = alpha + k","assumptions":["alpha, beta > 0","0 <= k <= n"],"name":"Beta-binomial conjugate posterior","equation":"conjugate update of the beta prior with the binomial likelihood"}

In your browser

Open Bayesian Statistics on ai999x.com, choose a capability and press Compute: the model runs on your own device as WebAssembly; nothing is sent anywhere. Change any input (any unit of the right kind) and the answer, its explanation and the figures follow.

In your own page: bayesian_statistics_ai999x.wasm and the loader ax-wasm.js (https://www.ai999x.com/compute/rust/ax-wasm.js).

Python

pip install bayesian-statistics-ai999x

Worked examples in every form

The same worked example from the command line, from Python and in the browser. Each command-line output is what the program printed.

Bayes factor for a normal mean (known variance)

BF10 for H1: mu ~ N(mu0, tau^2) against H0: mu = mu0, from the sample mean of n observations with known sigma; BF01 and the posterior probability of H1 at equal prior odds.

Command line:

$ bayesian-statistics solve bayes_factor_normal_mean sample_mean=0 n=1 sigma=1 mu0=0 prior_sd=1
bf10 = 0.7071067812
bf01 = 1.414213562
log10_bf10 = -0.1505149978
posterior_h1 = 0.4142135624
  by: Bayes factor for a normal mean (known variance)  (Savage-Dickey density ratio of the posterior and prior at mu0)
  sample_mean = 0
  sigma = 1
  mu0 = 0
  prior_sd = 1
  assumes: sigma, tau > 0; n >= 1

Python:

import bayesian_statistics_ai999x as m

a = m.solve("bayes_factor_normal_mean", sample_mean="0", n="1", sigma="1", mu0="0", prior_sd="1")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: This calculation determines the strength of evidence for a normal mean differing from a specified value using a known variance. By comparing the posterior and prior densities at the null value, it yields a Bayes factor of 0.707107 for the alternative model against the null. The corresponding inverse Bayes factor is 1.41421, and the logarithmic scale value is -0.150515. With equal prior odds, the posterior probability of the alternative model is estimated as 0.414214. These metrics rely on a sample size of 1, a sample mean of 0, and a prior standard deviation of 1, assuming a known population standard deviation of 1.

Bayes factor for a point null on a proportion (beta prior under H1)

BF10 for H1: p ~ Beta(alpha, beta) against H0: p = p0 after k successes in n trials (default uniform prior), with BF01 and the posterior probability of H1 at equal prior odds.

Command line:

$ bayesian-statistics solve bayes_factor_point_null_binomial p0=0.5 successes=2 trials=2 alpha=1 beta=1
bf10 = 1.333333333
bf01 = 0.75
log10_bf10 = 0.1249387366
posterior_h1 = 0.5714285714
  by: Bayes factor for a point null on a proportion (beta prior under H1)  (ratio of the beta-binomial marginal likelihood to the binomial likelihood at p0)
  p0 = 0.5
  alpha = 1
  beta = 1
  assumes: 0 < p0 < 1; alpha, beta > 0; 0 <= k <= n

Python:

import bayesian_statistics_ai999x as m

a = m.solve("bayes_factor_point_null_binomial", p0="0.5", successes="2", trials="2", alpha="1", beta="1")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: This calculation determines the strength of evidence for a beta-distributed proportion against a fixed null value. Given 2 successes in 2 trials, with a null hypothesis of 0.5 and prior parameters alpha of 1 and beta of 1, the resulting Bayes factor is 1.33333. The reciprocal, 0.75, represents evidence for the null. Under equal prior odds, the posterior probability that the alternative model is true is 0.571429.

Bayes factor for two simple binomial hypotheses

Bayes factor BF10 for H1: p = p1 against H0: p = p0 after k successes in n trials, its log10 and the posterior probability of H1 with equal prior odds.

Command line:

$ bayesian-statistics solve bayes_factor_binomial_simple p0=0.5 p1=0.7 successes=14 trials=20
bf10 = 5.184417905
log10_bf10 = 0.7147000018
posterior_h1 = 0.8383032946
  by: Bayes factor for two simple binomial hypotheses  (likelihood ratio of the two simple hypotheses)
  p0 = 0.5
  p1 = 0.7
  assumes: 0 < p0, p1 < 1; 0 <= k <= n

Python:

import bayesian_statistics_ai999x as m

a = m.solve("bayes_factor_binomial_simple", p0="0.5", p1="0.7", successes="14", trials="20")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: This calculation compares two simple hypotheses about a success probability using observed data consisting of 14 successes in 20 trials. The likelihood ratio yields a Bayes factor of 5.18442 for the hypothesis with probability 0.7 against the hypothesis with probability 0.5. This evidence translates to a log base ten scale value of 0.7147. Assuming equal prior odds between the two hypotheses, the resulting posterior probability that the first hypothesis is true given the data is 0.838303.

Marginal likelihood (evidence) of binomial data under a beta prior

Prior predictive probability of k successes in n trials when p ~ Beta(alpha, beta), and its logarithm.

Command line:

$ bayesian-statistics solve beta_binomial_evidence alpha=1 beta=1 successes=7 trials=10
marginal_likelihood = 0.09090909091
log_marginal_likelihood = -2.397895273
  by: Marginal likelihood (evidence) of binomial data under a beta prior  (closed form with log-beta functions)
  alpha = 1
  beta = 1
  assumes: alpha, beta > 0; 0 <= k <= n

Python:

import bayesian_statistics_ai999x as m

a = m.solve("beta_binomial_evidence", alpha="1", beta="1", successes="7", trials="10")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

What this means: This calculation determines the probability of observing exactly 7 successes out of 10 independent Bernoulli trials, assuming the underlying success rate follows a Beta distribution with parameters alpha equal to 1 and beta equal to 1. The resulting value, 0.0909091, represents the evidence for the data given the prior model. Its natural logarithm is computed as -2.3979, which is often more numerically stable for large numbers of trials.

Documentation

• Rust user guide: the command line, every capability with a command and what it printed: PDF · Markdown
• Python guide: every capability with a Python example: PDF · Markdown
• Desktop app & command line: downloads, install, ask and search, figures, offline use, checksums: https://www.ai999x.com/desktop-cli/
• Model page: compute in your browser, every capability explained: https://www.ai999x.com/models/bayesian-statistics-ai999x/

Highlights

• Verified, not generated. Every answer is computed by a certified bayesian statistics capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes (SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR), never a silent guess. Every one of its 378 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Native and in the browser. Every capability also computes natively in Rust: the desktop app and the bayesian-statistics command line for Windows and Linux, and WebAssembly on the model page, no Python, no network.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: Bayes' theorem, conjugate posteriors, credible intervals, Bayes factors, decision analysis, uncertainty propagation, Monte Carlo and sensitivity analysis.

Model overview

Property	Value
Package	bayesian-statistics-ai999x 1.5.0 (import bayesian_statistics_ai999x)
Type	Deterministic neuro-symbolic bayesian statistics engine (rule-based reading + certified symbolic capabilities)
Capabilities	78 certified procedures
Knowledge pack	version 1.0.1
Golden answers	378 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Python quick start

pip install bayesian-statistics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/bayesian-statistics-ai999x/resolve/main/dist/bayesian_statistics_ai999x-1.5.0-py3-none-any.whl

import bayesian_statistics_ai999x as bayesian_statistics

bayesian_statistics.ask("A test has sensitivity 0.9 and specificity 0.95; the prevalence is 0.01. What is the positive predictive value?").show()
# -> SOLVED — ppv = 0.153846, npv = 0.998938, p_positive = 0.0585, false_discovery_rate = 0.846154  (diagnostic_test_predictive_values)
bayesian_statistics.ask("Beta-binomial posterior: prior Beta(2, 2), 7 successes in 10 trials. Find the posterior mean.").show()
# -> SOLVED — post_alpha = 9, post_beta = 5, posterior_mean = 0.642857, posterior_variance = 0.0153061  (beta_binomial_posterior)
bayesian_statistics.ask("95% highest-density interval of a Beta(3, 9) posterior.").show()
# -> SOLVED — lower = 0.0405552, upper = 0.483724, width = 0.443168  (beta_hdi)
bayesian_statistics.ask("Expected value of perfect information: payoffs = [[100, -20], [40, 30]], probabilities = [0.4, 0.6].").show()
# -> SOLVED — evpi = 24, ev_with_perfect_information = 58, best_expected_value = 34  (expected_value_of_perfect_information)
bayesian_statistics.solve("uncertainty_product_quotient", a=12.0, ua=0.3, b=4.0, ub=0.2).show()
# -> SOLVED — product = 48, u_product = 2.68328, quotient = 3, u_quotient = 0.167705  (uncertainty_product_quotient)
bayesian_statistics.ask("Who created you?").message
# -> bayesian-statistics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

The Python package also installs a Python command, bayesian-statistics-ai999x "<question>"; for the native program use bayesian-statistics (above). The outputs above were produced by running this exact code on bayesian-statistics-ai999x 1.5.0.

How every answer is verified

1. Reading: the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning: only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing: exact symbolic or high-precision numeric evaluation.
4. Checking: dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Python worked examples, with their steps

Each question was put to bayesian-statistics-ai999x; the answer is the one it returned, with the steps it took.

Positive and negative predictive values of a test · Bayes' theorem and conditional probability

A test has sensitivity 0.9 and specificity 0.95; the prevalence is 0.01. What is the positive predictive value?

→ ppv = 0.1538461538, npv = 0.9989378651, p_positive = 0.0585, false_discovery_rate = 0.8461538462, false_omission_rate = 0.001062134891

Steps

• Positive and negative predictive values of a test: prevalence = 0.01, sensitivity = 0.9, specificity = 0.95 → ppv = 0.1538461538, npv = 0.9989378651, p_positive = 0.0585, false_discovery_rate = 0.8461538462, false_omission_rate = 0.001062134891

Beta-binomial conjugate posterior · Conjugate priors

Beta-binomial posterior: prior Beta(2, 2), 7 successes in 10 trials. Find the posterior mean.

→ post_alpha = 9, post_beta = 5, posterior_mean = 0.6428571429, posterior_variance = 0.01530612245, posterior_sd = 0.1237179148

Steps

• Beta-binomial conjugate posterior: alpha = 2, beta = 2, successes = 7, trials = 10 → post_alpha = 9, post_beta = 5, posterior_mean = 0.6428571429, posterior_variance = 0.01530612245, posterior_sd = 0.1237179148

Highest-density interval of a beta posterior · Credible intervals

95% highest-density interval of a Beta(3, 9) posterior.

→ lower = 0.04055516641, upper = 0.4837236593, width = 0.4431684929

Steps

• Highest-density interval of a beta posterior: alpha = 3, beta = 9, level = 0.95 → lower = 0.04055516641, upper = 0.4837236593, width = 0.4431684929

Expected value of perfect information (EVPI) · Decision analysis

Expected value of perfect information: payoffs = [[100, -20], [40, 30]], probabilities = [0.4, 0.6].

→ evpi = 24, ev_with_perfect_information = 58, best_expected_value = 34

Steps

• Expected value of perfect information (EVPI): payoffs = [100 -20; 40 30], probabilities = [0.4, 0.6] → evpi = 24, ev_with_perfect_information = 58, best_expected_value = 34

Uncertainty of a product and a quotient · Uncertainty propagation

m.solve("uncertainty_product_quotient", a=12.0, ua=0.3, b=4.0, ub=0.2)

→ product = 48, u_product = 2.683281573, quotient = 3, u_quotient = 0.1677050983

Steps

• Uncertainty of a product and a quotient: a = 12, ua = 0.3, b = 4, ub = 0.2 → product = 48, u_product = 2.683281573, quotient = 3, u_quotient = 0.1677050983

Bayes factor for two simple binomial hypotheses · Bayes factors and model comparison

Bayes factor for p1 = 0.7 against p0 = 0.5 after 14 successes in 20 trials.

→ bf10 = 5.184417905, log10_bf10 = 0.7147000018, posterior_h1 = 0.8383032946

Steps

• Bayes factor for two simple binomial hypotheses: p0 = 0.5, p1 = 0.7, successes = 14, trials = 20 → bf10 = 5.184417905, log10_bf10 = 0.7147000018, posterior_h1 = 0.8383032946

Figures

Every point was computed by bayesian-statistics-ai999x.

Beta prior and posterior after 7 successes in 10 trials

A Beta(2, 2) prior updated with 7 successes in 10 trials: the posterior Beta(9, 5) with mean 0.642857 and standard deviation 0.123718; shaded: the equal-tailed 95 % credible interval [0.385738, 0.861421]. Both densities are drawn from the package's cumulative probabilities at 199 values of θ (each interval's probability divided by its width).

Posterior mean for every count of successes and failures

The posterior mean of a success probability under a Beta(2, 2) prior after s successes (rows, 0 to 5) and f failures (columns, 0 to 5), each cell from the package (rounded to 3 decimals): from 0.222 after 5 failures to 0.778 after 5 successes, 0.5 with no data.

Probability of the condition after repeated tests

Repeated diagnostic tests with prevalence 0.01, sensitivity 0.9 and specificity 0.95: the package's posterior probability of the condition after k = 1 … 5 positive results (0.153846, 0.765957, 0.983308, 0.999058, 0.999948) and after 1 … 5 negative results (down to 1.3054e-07).

Prior and posterior over three hypotheses

Bayes' rule over 3 hypotheses with priors 0.5, 0.3, 0.2 and likelihoods 0.1, 0.4, 0.7: the package's posterior probabilities 0.16129, 0.387097, 0.451613 (evidence 0.31); the most probable hypothesis is 3 with 0.451613.

Category probabilities after multinomial counts

A Dirichlet(1, 1, 1) prior over 3 categories updated with the counts 10, 5, 3: the package's posterior Dirichlet(11, 6, 4), posterior means 0.52381, 0.285714, 0.190476; whiskers ±2 posterior standard deviations (0.106479, 0.0963143, 0.083719).

Normal posterior shrinking towards the data as n grows

A normal prior (mean 0, sd 1) and data with σ = 2 and sample mean 1: the package's posterior mean and 95 % credible interval for 17 sample sizes from 1 to 100, moving from the prior towards the data; marked: n = 10, posterior mean = 0.714286, interval [-0.333359, 1.76193].

Capabilities

78 certified procedures. A few of them:

• Procedures: Bayes action: minimum posterior expected loss; Bayes factor for a normal mean (known variance); Bayes factor for a point null on a proportion (beta prior under H1); Bayes factor for two simple binomial hypotheses; Bayes' theorem for an event and its complement; Bayesian A/B test: probability that A beats B; Best action by expected monetary value; Beta prior from a mean and standard deviation; Beta-binomial conjugate posterior; Break-even (indifference) probability of two actions; Certainty equivalent with logarithmic utility; Control-variate Monte Carlo estimate; Covariance propagation through a linear map y = A x; Credible interval for a Poisson rate; Credible interval for a proportion (beta-binomial); Decision threshold probability (treat / do not treat); Decision under uncertainty: maximin, maximax, Laplace, minimax regret, Hurwicz; Dirichlet-multinomial conjugate posterior; Dirichlet-multinomial probability of a count vector (evidence); Effective degrees of freedom and expanded uncertainty (Welch-Satterthwaite); Equal-tailed credible interval of a beta distribution; Equal-tailed credible interval of a gamma distribution; Exact mean and standard deviation of a product of independent normals; Exact mean and variance of a polynomial of a normal variable …

bayesian_statistics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 78 formulas

Show every formula, by topic

Bayes factors and model comparison

Bayes factor for a normal mean (known variance)

$$B{F}_{01}=\frac{p({\mu }_0\mid \bar{x},H_1)}{p({\mu }_0\mid H_1)}BF\_\{01\}\; =\; \backslash frac\{p(\backslash mu\_0\backslash mid\; \backslash bar\; x,\; H\_1)\}\{p(\backslash mu\_0\backslash mid\; H\_1)\}$$

Bayes factor for a point null on a proportion (beta prior under H1)

$$B{F}_{10}=\frac{B(\alpha +k,\beta +n-k)\mathrm{/}B(\alpha ,\beta )}{p_0^k(1-p_0{)}^{n-k}}BF\_\{10\}\; =\; \backslash frac\{B(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k)/B(\backslash alpha,\backslash beta)\}\{p\_0^\{k\}(1-p\_0)^\{n-k\}\}$$

Bayes factor for two simple binomial hypotheses

$$B{F}_{10}={\left(\frac{p_1}{p_0}\right)}^k{\left(\frac{1-p_1}{1-p_0}\right)}^{n-k}BF\_\{10\}\; =\; \backslash left(\backslash frac\{p\_1\}\{p\_0\}\backslash right)^\{k\}\backslash left(\backslash frac\{1-p\_1\}\{1-p\_0\}\backslash right)^\{n-k\}$$

Marginal likelihood (evidence) of binomial data under a beta prior

$$p(k)=\left(\genfrac{}{}{0px}{}{n}{k}\right)\frac{B(\alpha +k,\beta +n-k)}{B(\alpha ,\beta )}p(k)\; =\; \backslash binom\{n\}\{k\}\backslash frac\{B(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k)\}\{B(\backslash alpha,\backslash beta)\}$$

Posterior model probabilities from log evidences

$$P(M_i\mid D)=\frac{P(D\mid M_i)P(M_i)}{\sum _{j}P(D\mid M_j)P(M_j)}P(M\_i\backslash mid\; D)\; =\; \backslash frac\{P(D\backslash mid\; M\_i)P(M\_i)\}\{\backslash sum\_j\; P(D\backslash mid\; M\_j)P(M\_j)\}$$

Posterior probability of a hypothesis from a Bayes factor

$$\frac{P(H_1\mid D)}{P(H_0\mid D)}=B{F}_{10}\frac{P(H_1)}{P(H_0)}\backslash frac\{P(H\_1\backslash mid\; D)\}\{P(H\_0\backslash mid\; D)\}\; =\; BF\_\{10\}\backslash ,\backslash frac\{P(H\_1)\}\{P(H\_0)\}$$

Bayes' theorem and conditional probability

Bayes' theorem for an event and its complement

$$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B\mid A)P(A)+P(B\mid \mathrm{\neg }A)(1-P(A))}P(A\backslash mid\; B)\; =\; \backslash frac\{P(B\backslash mid\; A)\backslash ,P(A)\}\{P(B\backslash mid\; A)\backslash ,P(A)\; +\; P(B\backslash mid\; \backslash neg\; A)\backslash ,(1\; -\; P(A))\}$$

Law of total probability

$$P(B)=\sum _{i}P(B\mid A_i)P(A_i)P(B)\; =\; \backslash sum\_i\; P(B\backslash mid\; A\_i)\backslash ,P(A\_i)$$

Positive and negative predictive values of a test

$$\mathrm{P}\mathrm{P}\mathrm{V}=\frac{\mathrm{S}\mathrm{e}\pi }{\mathrm{S}\mathrm{e}\pi +(1-\mathrm{S}\mathrm{p})(1-\pi )},\mathrm{N}\mathrm{P}\mathrm{V}=\frac{\mathrm{S}\mathrm{p}(1-\pi )}{\mathrm{S}\mathrm{p}(1-\pi )+(1-\mathrm{S}\mathrm{e})\pi }\backslash mathrm\{PPV\}\; =\; \backslash frac\{\backslash mathrm\{Se\}\backslash ,\backslash pi\}\{\backslash mathrm\{Se\}\backslash ,\backslash pi\; +\; (1-\backslash mathrm\{Sp\})(1-\backslash pi)\},\backslash quad\; \backslash mathrm\{NPV\}\; =\; \backslash frac\{\backslash mathrm\{Sp\}\backslash ,(1-\backslash pi)\}\{\backslash mathrm\{Sp\}\backslash ,(1-\backslash pi)\; +\; (1-\backslash mathrm\{Se\})\backslash ,\backslash pi\}$$

Posterior after repeated independent tests

$${\text{odds}}_{post}=\frac{\pi }{1-\pi }(L{R}^{+}{)}^k(L{R}^{-}{)}^m\backslash text\{odds\}\_\{post\}\; =\; \backslash frac\{\backslash pi\}\{1-\backslash pi\}\backslash ,(LR^\{+\})^\{k\}\backslash ,(LR^\{-\})^\{m\}$$

Posterior over several discrete hypotheses

$$P(H_i\mid D)=\frac{P(D\mid H_i)P(H_i)}{\sum _{j}P(D\mid H_j)P(H_j)}P(H\_i\backslash mid\; D)\; =\; \backslash frac\{P(D\backslash mid\; H\_i)\backslash ,P(H\_i)\}\{\backslash sum\_j\; P(D\backslash mid\; H\_j)\backslash ,P(H\_j)\}$$

Monte Carlo estimation

Control-variate Monte Carlo estimate

$${\widehat{\mu }}_{cv}=\bar{y}-\widehat{b}(\bar{c}-{\mu }_c),\widehat{b}=\frac{\sum (y_i-\bar{y})(c_i-\bar{c})}{\sum (c_i-\bar{c}{)}^2}\backslash hat\backslash mu\_\{cv\}\; =\; \backslash bar\; y\; -\; \backslash hat\; b\backslash ,(\backslash bar\; c\; -\; \backslash mu\_c),\backslash quad\; \backslash hat\; b\; =\; \backslash frac\{\backslash sum\; (y\_i-\backslash bar\; y)(c\_i-\backslash bar\; c)\}\{\backslash sum\; (c\_i-\backslash bar\; c)^2\}$$

Importance sampling estimate and effective sample size

$$\widehat{\mu }=\frac{\sum _i{w}_{i}f(x_i)}{\sum _i{w}_i},w_i=\frac{p(x_i)}{q(x_i)},ESS=\frac{(\sum w_i{)}^2}{\sum w_i^2}\backslash hat\backslash mu\; =\; \backslash frac\{\backslash sum\_i\; w\_i\; f(x\_i)\}\{\backslash sum\_i\; w\_i\},\backslash \; w\_i\; =\; \backslash frac\{p(x\_i)\}\{q(x\_i)\},\backslash quad\; ESS\; =\; \backslash frac\{(\backslash sum\; w\_i)^2\}\{\backslash sum\; w\_i^2\}$$

Monte Carlo estimate of an integral from uniform samples

$$\widehat{I}=\frac{b-a}{n}\sum _{i}f(x_i),SE=(b-a)\frac{s_f}{\sqrt{n}}\backslash hat\; I\; =\; \backslash frac\{b-a\}\{n\}\backslash sum\_i\; f(x\_i),\backslash quad\; SE\; =\; (b-a)\backslash frac\{s\_f\}\{\backslash sqrt\; n\}$$

Number of Monte Carlo draws for a target accuracy

$$n=\lceil {\left(\frac{z_{(1+c)\mathrm{/}2}\sigma }{E}\right)}^2\rceil n\; =\; \backslash left\backslash lceil\backslash left(\backslash frac\{z\_\{(1+c)/2\}\backslash ,\backslash sigma\}\{E\}\backslash right)^2\backslash right\backslash rceil$$

Point of the Halton low-discrepancy sequence

$${\varphi }_b(i)=\sum _k{d}_k{b}^{-k-1}\text{ for }i=\sum _k{d}_k{b}^k\backslash phi\_b(i)\; =\; \backslash sum\_k\; d\_k\backslash ,b^\{-k-1\}\backslash \; \backslash text\{\; for\; \}\; i\; =\; \backslash sum\_k\; d\_k\; b^k$$

Stratified sampling estimate and standard error

$$\widehat{\mu }=\sum _h{W}_h{\bar{y}}_h,SE=\sqrt{\sum _h{W}_h^2{s}_h^2\mathrm{/}{n}_h}\backslash hat\backslash mu\; =\; \backslash sum\_h\; W\_h\backslash bar\; y\_h,\backslash quad\; SE\; =\; \backslash sqrt\{\backslash sum\_h\; W\_h^2\; s\_h^2/n\_h\}$$

Conjugate priors: Dirichlet-multinomial

Dirichlet-multinomial conjugate posterior

$$\theta \mid n\sim \mathrm{D}\mathrm{i}\mathrm{r}({\alpha }_1+n_1,\dots ,{\alpha }_k+n_k),E[{\theta }_j]=\frac{{\alpha }_j+n_j}{{\alpha }_0+n}\backslash theta\backslash mid\; n\; \backslash sim\; \backslash mathrm\{Dir\}(\backslash alpha\_1\; +\; n\_1,\; \backslash ldots,\; \backslash alpha\_k\; +\; n\_k),\backslash quad\; E[\backslash theta\_j]\; =\; \backslash frac\{\backslash alpha\_j\; +\; n\_j\}\{\backslash alpha\_0\; +\; n\}$$

Dirichlet-multinomial probability of a count vector (evidence)

$$P(n\mid \alpha )=\frac{n!\mathrm{\Gamma }({\alpha }_0)}{\mathrm{\Gamma }(n+{\alpha }_0)}\prod _j\frac{\mathrm{\Gamma }(n_j+{\alpha }_j)}{n_j!\mathrm{\Gamma }({\alpha }_j)}P(n\backslash mid\backslash alpha)\; =\; \backslash frac\{n!\backslash ,\backslash Gamma(\backslash alpha\_0)\}\{\backslash Gamma(n\; +\; \backslash alpha\_0)\}\backslash prod\_j\backslash frac\{\backslash Gamma(n\_j\; +\; \backslash alpha\_j)\}\{n\_j!\backslash ,\backslash Gamma(\backslash alpha\_j)\}$$

Conjugate priors: beta-binomial

Bayesian A/B test: probability that A beats B

$$P(p_B>p_A)=\sum _{i=0}^{{\alpha }_B-1}\frac{B({\alpha }_A+i,{\beta }_A+{\beta }_B)}{({\beta }_B+i)B(1+i,{\beta }_B)B({\alpha }_A,{\beta }_A)}P(p\_B\; >\; p\_A)\; =\; \backslash sum\_\{i=0\}^\{\backslash alpha\_B\; -\; 1\}\backslash frac\{B(\backslash alpha\_A\; +\; i,\backslash \; \backslash beta\_A\; +\; \backslash beta\_B)\}\{(\backslash beta\_B\; +\; i)\backslash ,B(1\; +\; i,\backslash \; \backslash beta\_B)\backslash ,B(\backslash alpha\_A,\; \backslash beta\_A)\}$$

Beta-binomial conjugate posterior

$$p\mid k\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha +k,\beta +n-k),E[p\mid k]=\frac{\alpha +k}{\alpha +\beta +n}p\backslash mid\; k\; \backslash sim\; \backslash mathrm\{Beta\}(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k),\backslash quad\; E[p\backslash mid\; k]\; =\; \backslash frac\{\backslash alpha\; +\; k\}\{\backslash alpha\; +\; \backslash beta\; +\; n\}$$

Laplace's rule of succession

$$P(\text{next success})=\frac{k+1}{n+2}P(\backslash text\{next\; success\})\; =\; \backslash frac\{k+1\}\{n+2\}$$

MAP estimate of a proportion (beta-binomial)

$${\widehat{p}}_{MAP}=\frac{\alpha +k-1}{\alpha +\beta +n-2}\backslash hat\; p\_\{MAP\}\; =\; \backslash frac\{\backslash alpha\; +\; k\; -\; 1\}\{\backslash alpha\; +\; \backslash beta\; +\; n\; -\; 2\}$$

Posterior predictive probability (beta-binomial)

$$P(J=j)=\left(\genfrac{}{}{0px}{}{m}{j}\right)\frac{B(\alpha +j,\beta +m-j)}{B(\alpha ,\beta )}P(J\; =\; j)\; =\; \backslash binom\{m\}\{j\}\backslash frac\{B(\backslash alpha\; +\; j,\backslash \; \backslash beta\; +\; m\; -\; j)\}\{B(\backslash alpha,\; \backslash beta)\}$$

Posterior probability that p exceeds a value

$$P(p>c\mid k)=1-I_c(\alpha +k,\beta +n-k)P(p\; >\; c\; \backslash mid\; k)\; =\; 1\; -\; I\_c(\backslash alpha\; +\; k,\backslash \; \backslash beta\; +\; n\; -\; k)$$

Conjugate priors: gamma-Poisson and gamma-exponential

Gamma-Poisson conjugate posterior of a rate

$$\lambda \mid x\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\text{\hspace{0.17em}⁣}(\alpha +\sum x_i,\beta +n)\backslash lambda\backslash mid\; x\; \backslash sim\; \backslash mathrm\{Gamma\}\backslash !\backslash left(\backslash alpha\; +\; \backslash textstyle\backslash sum\; x\_i,\backslash \; \backslash beta\; +\; n\backslash right)$$

Gamma-exponential conjugate posterior of a rate

$$\lambda \mid t\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\text{\hspace{0.17em}⁣}(\alpha +n,\beta +\sum t_i)\backslash lambda\backslash mid\; t\; \backslash sim\; \backslash mathrm\{Gamma\}\backslash !\backslash left(\backslash alpha\; +\; n,\backslash \; \backslash beta\; +\; \backslash textstyle\backslash sum\; t\_i\backslash right)$$

MAP estimate of a Poisson rate (gamma prior)

$${\widehat{\lambda }}_{MAP}=\frac{\alpha +\sum x_i-1}{\beta +n}\backslash hat\backslash lambda\_\{MAP\}\; =\; \backslash frac\{\backslash alpha\; +\; \backslash sum\; x\_i\; -\; 1\}\{\backslash beta\; +\; n\}$$

Posterior predictive count (negative binomial)

$$P(X=k)=\frac{\mathrm{\Gamma }(\alpha +k)}{\mathrm{\Gamma }(\alpha )k!}{\left(\frac{\beta }{\beta +1}\right)}^{\alpha }{\left(\frac{1}{\beta +1}\right)}^{k}P(X\; =\; k)\; =\; \backslash frac\{\backslash Gamma(\backslash alpha\; +\; k)\}\{\backslash Gamma(\backslash alpha)\backslash ,k!\}\backslash left(\backslash frac\{\backslash beta\}\{\backslash beta+1\}\backslash right)^\{\backslash alpha\}\backslash left(\backslash frac\{1\}\{\backslash beta+1\}\backslash right)^\{k\}$$

Probability that one Poisson rate exceeds another (gamma posteriors)

$$P({\lambda }_1>{\lambda }_2)=1-I_{b_1\mathrm{/}(b_1+b_2)}(a_1,a_2)P(\backslash lambda\_1\; >\; \backslash lambda\_2)\; =\; 1\; -\; I\_\{b\_1/(b\_1+b\_2)\}(a\_1,\; a\_2)$$

Conjugate priors: normal models

Normal mean with known variance: posterior and credible interval from a sample mean

$${\mu }_n=B{\mu }_0+(1-B)\bar{x},B=\frac{{\sigma }^2\mathrm{/}n}{{\sigma }^2\mathrm{/}n+{\tau }_0^2},{\mu }_n\pm z_{(1+c)\mathrm{/}2}{\tau }_n\backslash mu\_n\; =\; B\backslash mu\_0\; +\; (1-B)\backslash bar\; x,\backslash \; B\; =\; \backslash frac\{\backslash sigma^2/n\}\{\backslash sigma^2/n\; +\; \backslash tau\_0^2\},\backslash quad\; \backslash mu\_n\; \backslash pm\; z\_\{(1+c)/2\}\backslash ,\backslash tau\_n$$

Normal mean with known variance: posterior from data

$$\frac{1}{{\tau }_n^2}=\frac{1}{{\tau }_0^2}+\frac{n}{{\sigma }^2},{\mu }_n={\tau }_n^2(\frac{{\mu }_0}{{\tau }_0^2}+\frac{n\bar{x}}{{\sigma }^2})\backslash frac\{1\}\{\backslash tau\_n^2\}\; =\; \backslash frac\{1\}\{\backslash tau\_0^2\}\; +\; \backslash frac\{n\}\{\backslash sigma^2\},\backslash quad\; \backslash mu\_n\; =\; \backslash tau\_n^2\backslash left(\backslash frac\{\backslash mu\_0\}\{\backslash tau\_0^2\}\; +\; \backslash frac\{n\backslash bar\; x\}\{\backslash sigma^2\}\backslash right)$$

Normal model with unknown mean and variance (normal-inverse-gamma)

$${\beta }_n={\beta }_0+\frac{1}{2}\sum (x_i-\bar{x}{)}^2+\frac{{\kappa }_{0}n(\bar{x}-{\mu }_0{)}^2}{2({\kappa }_0+n)},\mu \mid x\sim t_{2{\alpha }_n}\text{\hspace{0.17em}⁣}({\mu }_n,\frac{{\beta }_n}{{\alpha }_n{\kappa }_n})\backslash beta\_n\; =\; \backslash beta\_0\; +\; \backslash tfrac12\backslash sum(x\_i-\backslash bar\; x)^2\; +\; \backslash frac\{\backslash kappa\_0\; n(\backslash bar\; x-\backslash mu\_0)^2\}\{2(\backslash kappa\_0+n)\},\backslash quad\; \backslash mu\backslash mid\; x\; \backslash sim\; t\_\{2\backslash alpha\_n\}\backslash !\backslash left(\backslash mu\_n,\backslash \; \backslash frac\{\backslash beta\_n\}\{\backslash alpha\_n\backslash kappa\_n\}\backslash right)$$

Posterior predictive distribution of a new observation

$$Y_{new}\mid \bar{x}\sim N\text{\hspace{0.17em}⁣}({\mu }_n,{\tau }_n^2+{\sigma }^2)Y\_\{new\}\backslash mid\; \backslash bar\; x\; \backslash sim\; N\backslash !\backslash left(\backslash mu\_n,\backslash \; \backslash tau\_n^2\; +\; \backslash sigma^2\backslash right)$$

Credible intervals

Credible interval for a Poisson rate

$$[\frac{G_{{\alpha }^{\mathrm{\prime }}}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1-c}{2}\right)}{{\beta }^{\mathrm{\prime }}},\frac{G_{{\alpha }^{\mathrm{\prime }}}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1+c}{2}\right)}{{\beta }^{\mathrm{\prime }}}]\backslash left[\backslash frac\{G^\{-1\}\_\{\backslash alpha\text{'}\}\backslash !\backslash left(\backslash frac\{1-c\}\{2\}\backslash right)\}\{\backslash beta\text{'}\},\backslash \; \backslash frac\{G^\{-1\}\_\{\backslash alpha\text{'}\}\backslash !\backslash left(\backslash frac\{1+c\}\{2\}\backslash right)\}\{\backslash beta\text{'}\}\backslash right]$$

Credible interval for a proportion (beta-binomial)

$$p\mid k\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha +k,\beta +n-k),[q_{(1-c)\mathrm{/}2},q_{(1+c)\mathrm{/}2}]p\backslash mid\; k\; \backslash sim\; \backslash mathrm\{Beta\}(\backslash alpha+k,\backslash beta+n-k),\backslash quad\; [q\_\{(1-c)/2\},\backslash \; q\_\{(1+c)/2\}]$$

Equal-tailed credible interval of a beta distribution

$$[F_{\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha ,\beta )}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1-c}{2}\right),F_{\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\alpha ,\beta )}^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1+c}{2}\right)]\backslash left[F^\{-1\}\_\{\backslash mathrm\{Beta\}(\backslash alpha,\backslash beta)\}\backslash !\backslash left(\backslash tfrac\{1-c\}\{2\}\backslash right),\backslash \; F^\{-1\}\_\{\backslash mathrm\{Beta\}(\backslash alpha,\backslash beta)\}\backslash !\backslash left(\backslash tfrac\{1+c\}\{2\}\backslash right)\backslash right]$$

Equal-tailed credible interval of a gamma distribution

$$[G^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1-c}{2}\right),G^{-1}\text{\hspace{0.17em}⁣}\left(\frac{1+c}{2}\right)]\backslash left[G^\{-1\}\backslash !\backslash left(\backslash tfrac\{1-c\}\{2\}\backslash right),\backslash \; G^\{-1\}\backslash !\backslash left(\backslash tfrac\{1+c\}\{2\}\backslash right)\backslash right]$$

Highest-density interval of a beta posterior

$$F(u)-F(l)=c,f(l)=f(u)F(u)\; -\; F(l)\; =\; c,\backslash quad\; f(l)\; =\; f(u)$$

Highest-density interval of a gamma posterior

$$G(u)-G(l)=c,g(l)=g(u)G(u)\; -\; G(l)\; =\; c,\backslash quad\; g(l)\; =\; g(u)$$

Jeffreys interval for a binomial proportion

$$[B^{-1}\text{\hspace{0.17em}⁣}(\frac{1-c}{2};k+\frac{1}{2},n-k+\frac{1}{2}),B^{-1}\text{\hspace{0.17em}⁣}(\frac{1+c}{2};k+\frac{1}{2},n-k+\frac{1}{2})]\backslash left[B^\{-1\}\backslash !\backslash left(\backslash tfrac\{1-c\}\{2\};\backslash ,k+\backslash tfrac12,\backslash ,n-k+\backslash tfrac12\backslash right),\backslash \; B^\{-1\}\backslash !\backslash left(\backslash tfrac\{1+c\}\{2\};\backslash ,k+\backslash tfrac12,\backslash ,n-k+\backslash tfrac12\backslash right)\backslash right]$$

Decision analysis

Bayes action: minimum posterior expected loss

$$\rho (a)=\sum _{s}P(s\mid x)L(a,s),a^{\ast }=\arg \underset{a}{\min }\rho (a)\backslash rho(a)\; =\; \backslash sum\_s\; P(s\backslash mid\; x)\backslash ,L(a,s),\backslash quad\; a^*\; =\; \backslash arg\backslash min\_a\; \backslash rho(a)$$

Best action by expected monetary value

$$EV(a)=\sum _{s}P(s)u(a,s),a^{\ast }=\arg \underset{a}{\max }EV(a)EV(a)\; =\; \backslash sum\_s\; P(s)\backslash ,u(a,\; s),\backslash quad\; a^*\; =\; \backslash arg\backslash max\_a\; EV(a)$$

Break-even (indifference) probability of two actions

$$p^{\ast }=\frac{u_{22}-u_{12}}{(u_{11}-u_{12})-(u_{21}-u_{22})}p^*\; =\; \backslash frac\{u\_\{22\}\; -\; u\_\{12\}\}\{(u\_\{11\}\; -\; u\_\{12\})\; -\; (u\_\{21\}\; -\; u\_\{22\})\}$$

Certainty equivalent with logarithmic utility

$$CE=\exp \text{\hspace{0.17em}⁣}(\sum _i{p}_i\ln x_i)=\prod _i{x}_i^{p_i}CE\; =\; \backslash exp\backslash !\backslash left(\backslash sum\_i\; p\_i\; \backslash ln\; x\_i\backslash right)\; =\; \backslash prod\_i\; x\_i^\{p\_i\}$$

Decision threshold probability (treat / do not treat)

$$p^{\ast }=\frac{L_{FP}}{L_{FP}+L_{FN}}p^*\; =\; \backslash frac\{L\_\{FP\}\}\{L\_\{FP\}\; +\; L\_\{FN\}\}$$

Decision under uncertainty: maximin, maximax, Laplace, minimax regret, Hurwicz

$$\text{maximin: }\underset{a}{\max }\underset{s}{\min }u,\text{regret }r(a,s)=\underset{a^{\mathrm{\prime }}}{\max }u(a^{\mathrm{\prime }},s)-u(a,s)\backslash text\{maximin:\; \}\backslash max\_a\backslash min\_s\; u,\backslash quad\; \backslash text\{regret\; \}\; r(a,s)\; =\; \backslash max\_\{a\text{'}\}u(a\text{'},s)\; -\; u(a,s)$$

Expected utility and certainty equivalent (exponential utility)

$$CE=-R\ln \sum _i{p}_i{e}^{-x_i\mathrm{/}R}CE\; =\; -R\backslash ln\backslash sum\_i\; p\_i\; e^\{-x\_i/R\}$$

Expected value of perfect information (EVPI)

$$EVPI=\sum _{s}P(s)\underset{a}{\max }u(a,s)-\underset{a}{\max }\sum _{s}P(s)u(a,s)EVPI\; =\; \backslash sum\_s\; P(s)\backslash max\_a\; u(a,s)\; -\; \backslash max\_a\backslash sum\_s\; P(s)\backslash ,u(a,s)$$

Expected value of sample information (EVSI)

$$EVSI=\sum _{z}P(z)\underset{a}{\max }\sum _{s}P(s\mid z)u(a,s)-\underset{a}{\max }\sum _{s}P(s)u(a,s)EVSI\; =\; \backslash sum\_z\; P(z)\backslash max\_a\backslash sum\_s\; P(s\backslash mid\; z)\backslash ,u(a,s)\; -\; \backslash max\_a\backslash sum\_s\; P(s)\backslash ,u(a,s)$$

Information and quadrature for Bayesian computation

Gauss-Hermite rule for expectations under a normal distribution

$$E[f(Z)]\approx \sum _{i=1}^n{w}_{i}f(x_i),H{e}_n(x_i)=0,w_i=\frac{n!}{n^{2}H{e}_{n-1}(x_i{)}^2}E[f(Z)]\; \backslash approx\; \backslash sum\_\{i=1\}^\{n\}\; w\_i\; f(x\_i),\backslash quad\; He\_n(x\_i)\; =\; 0,\backslash quad\; w\_i\; =\; \backslash frac\{n!\}\{n^2\backslash ,He\_\{n-1\}(x\_i)^2\}$$

Kullback-Leibler divergence between two beta distributions

$$KL=\ln \frac{B({\alpha }_2,{\beta }_2)}{B({\alpha }_1,{\beta }_1)}+({\alpha }_1-{\alpha }_2)\psi ({\alpha }_1)+({\beta }_1-{\beta }_2)\psi ({\beta }_1)+({\alpha }_2-{\alpha }_1+{\beta }_2-{\beta }_1)\psi ({\alpha }_1+{\beta }_1)KL\; =\; \backslash ln\backslash frac\{B(\backslash alpha\_2,\backslash beta\_2)\}\{B(\backslash alpha\_1,\backslash beta\_1)\}\; +\; (\backslash alpha\_1-\backslash alpha\_2)\backslash psi(\backslash alpha\_1)\; +\; (\backslash beta\_1-\backslash beta\_2)\backslash psi(\backslash beta\_1)\; +\; (\backslash alpha\_2-\backslash alpha\_1+\backslash beta\_2-\backslash beta\_1)\backslash psi(\backslash alpha\_1+\backslash beta\_1)$$

Kullback-Leibler divergence between two normal distributions

$$KL(N_1\mathrm{\parallel }{N}_2)=\ln \frac{{\sigma }_2}{{\sigma }_1}+\frac{{\sigma }_1^2+({\mu }_1-{\mu }_2{)}^2}{2{\sigma }_2^2}-\frac{1}{2}KL(N\_1\backslash Vert\; N\_2)\; =\; \backslash ln\backslash frac\{\backslash sigma\_2\}\{\backslash sigma\_1\}\; +\; \backslash frac\{\backslash sigma\_1^2\; +\; (\backslash mu\_1-\backslash mu\_2)^2\}\{2\backslash sigma\_2^2\}\; -\; \backslash frac12$$

Odds and likelihood ratios

Likelihood ratios of a diagnostic test

$$L{R}^{+}=\frac{\mathrm{S}\mathrm{e}}{1-\mathrm{S}\mathrm{p}},L{R}^{-}=\frac{1-\mathrm{S}\mathrm{e}}{\mathrm{S}\mathrm{p}}LR^\{+\}\; =\; \backslash frac\{\backslash mathrm\{Se\}\}\{1-\backslash mathrm\{Sp\}\},\backslash quad\; LR^\{-\}\; =\; \backslash frac\{1-\backslash mathrm\{Se\}\}\{\backslash mathrm\{Sp\}\}$$

Odds and log-odds of a probability

$$\text{odds}=\frac{p}{1-p},\mathrm{logit}p=\ln \frac{p}{1-p}\backslash text\{odds\}\; =\; \backslash frac\{p\}\{1-p\},\backslash quad\; \backslash operatorname\{logit\}\; p\; =\; \backslash ln\backslash frac\{p\}\{1-p\}$$

Post-test probability from a likelihood ratio

$$\frac{P(H\mid E)}{1-P(H\mid E)}=\frac{P(H)}{1-P(H)}\times LR\backslash frac\{P(H\backslash mid\; E)\}\{1-P(H\backslash mid\; E)\}\; =\; \backslash frac\{P(H)\}\{1-P(H)\}\backslash times\; LR$$

Probability from odds

$$p=\frac{o}{1+o}p\; =\; \backslash frac\{o\}\{1+o\}$$

Posterior approximation

Laplace (normal) approximation of a beta posterior

$$\widehat{p}=\frac{\alpha -1}{\alpha +\beta -2},{\sigma }^2={[\frac{\alpha -1}{{\widehat{p}}^2}+\frac{\beta -1}{(1-\widehat{p}{)}^2}]}^{-1}\backslash hat\; p\; =\; \backslash frac\{\backslash alpha-1\}\{\backslash alpha+\backslash beta-2\},\backslash quad\; \backslash sigma^2\; =\; \backslash left[\backslash frac\{\backslash alpha-1\}\{\backslash hat\; p^2\}\; +\; \backslash frac\{\backslash beta-1\}\{(1-\backslash hat\; p)^2\}\backslash right]^\{-1\}$$

Prior elicitation

Beta prior from a mean and standard deviation

$$\nu =\frac{m(1-m)}{s^2}-1,\alpha =m\nu ,\beta =(1-m)\nu \backslash nu\; =\; \backslash frac\{m(1-m)\}\{s^2\}\; -\; 1,\backslash quad\; \backslash alpha\; =\; m\backslash nu,\backslash quad\; \backslash beta\; =\; (1-m)\backslash nu$$

Sensitivity analysis

Morris elementary-effects screening

$$E{E}_j=\frac{f(x+\mathrm{\Delta }{e}_j)-f(x)}{\mathrm{\Delta }},{\mu }_j^{\ast }=\frac{1}{r}\sum _t\mathrm{\mid }E{E}_j^{(t)}\mathrm{\mid }EE\_j\; =\; \backslash frac\{f(x\; +\; \backslash Delta\; e\_j)\; -\; f(x)\}\{\backslash Delta\},\backslash quad\; \backslash mu\_j^*\; =\; \backslash frac1r\backslash sum\_t\; \backslash vert\; EE\_j^\{(t)\}\backslash vert$$

Sobol indices of Sobol's G-function

$$V_i=\frac{1}{3(1+a_i{)}^2},V=\prod _i(1+V_i)-1,S_i=\frac{V_i}{V},S_{Ti}=\frac{V_i\prod _{j\mathrm{\ne }i}(1+V_j)}{V}V\_i\; =\; \backslash frac\{1\}\{3(1+a\_i)^2\},\backslash \; V\; =\; \backslash prod\_i(1+V\_i)\; -\; 1,\backslash \; S\_i\; =\; \backslash frac\{V\_i\}\{V\},\backslash \; S\_\{Ti\}\; =\; \backslash frac\{V\_i\backslash prod\_\{j\backslash ne\; i\}(1+V\_j)\}\{V\}$$

Sobol indices of a linear (additive) model

$$S_i=\frac{a_i^2{\sigma }_i^2}{\sum _j{a}_j^2{\sigma }_j^2}S\_i\; =\; \backslash frac\{a\_i^2\backslash sigma\_i^2\}\{\backslash sum\_j\; a\_j^2\backslash sigma\_j^2\}$$

Sobol indices of a model with an interaction term

$$V_1=(a_1+a_{12}{\mu }_2{)}^2{\sigma }_1^2,V_2=(a_2+a_{12}{\mu }_1{)}^2{\sigma }_2^2,V_{12}=a_{12}^2{\sigma }_1^2{\sigma }_2^{2}V\_1\; =\; (a\_1\; +\; a\_\{12\}\backslash mu\_2)^2\backslash sigma\_1^2,\backslash \; V\_2\; =\; (a\_2\; +\; a\_\{12\}\backslash mu\_1)^2\backslash sigma\_2^2,\backslash \; V\_\{12\}\; =\; a\_\{12\}^2\backslash sigma\_1^2\backslash sigma\_2^2$$

Sobol sensitivity indices of the Ishigami function

$$V=\frac{a^2}{8}+\frac{b{\pi }^4}{5}+\frac{b^2{\pi }^8}{18}+\frac{1}{2},V_1=\frac{1}{2}{(1+\frac{b{\pi }^4}{5})}^2,V_2=\frac{a^2}{8},V_{13}=\frac{8b^2{\pi }^8}{225}V\; =\; \backslash frac\{a^2\}\{8\}\; +\; \backslash frac\{b\backslash pi^4\}\{5\}\; +\; \backslash frac\{b^2\backslash pi^8\}\{18\}\; +\; \backslash frac12,\backslash \; V\_1\; =\; \backslash frac12\backslash left(1\; +\; \backslash frac\{b\backslash pi^4\}\{5\}\backslash right)^2,\backslash \; V\_2\; =\; \backslash frac\{a^2\}\{8\},\backslash \; V\_\{13\}\; =\; \backslash frac\{8b^2\backslash pi^8\}\{225\}$$

Uncertainty propagation

Covariance propagation through a linear map y = A x

$${\mathrm{\Sigma }}_y=A{\mathrm{\Sigma }}_x{A}^{\mathsf{T}}\backslash Sigma\_y\; =\; A\backslash ,\backslash Sigma\_x\backslash ,A^\{\backslash mathsf\; T\}$$

Effective degrees of freedom and expanded uncertainty (Welch-Satterthwaite)

$${\nu }_{eff}=\frac{u_c^4}{\sum _i{u}_i^4\mathrm{/}{\nu }_i},U=t_{(1+p)\mathrm{/}2,{\nu }_{eff}}{u}_c\backslash nu\_\{eff\}\; =\; \backslash frac\{u\_c^4\}\{\backslash sum\_i\; u\_i^4/\backslash nu\_i\},\backslash quad\; U\; =\; t\_\{(1+p)/2,\backslash ,\backslash nu\_\{eff\}\}\backslash ,u\_c$$

Exact mean and standard deviation of a product of independent normals

$$\mathrm{V}\mathrm{a}\mathrm{r}(XY)={\mu }_x^2{\sigma }_y^2+{\mu }_y^2{\sigma }_x^2+{\sigma }_x^2{\sigma }_y^2\backslash mathrm\{Var\}(XY)\; =\; \backslash mu\_x^2\backslash sigma\_y^2\; +\; \backslash mu\_y^2\backslash sigma\_x^2\; +\; \backslash sigma\_x^2\backslash sigma\_y^2$$

Exact mean and variance of a polynomial of a normal variable

$$E[Z^{2m}]=(2m-1)!!,X=\mu +\sigma ZE[Z^\{2m\}]\; =\; (2m-1)!!,\backslash quad\; X\; =\; \backslash mu\; +\; \backslash sigma\; Z$$

Exact versus first-order propagation through exp(X)

$$E[e^X]=e^{\mu +{\sigma }^2\mathrm{/}2},\mathrm{S}\mathrm{D}[e^X]=e^{\mu +{\sigma }^2\mathrm{/}2}\sqrt{e^{{\sigma }^2}-1}E[e^X]\; =\; e^\{\backslash mu\; +\; \backslash sigma^2/2\},\backslash quad\; \backslash mathrm\{SD\}[e^X]\; =\; e^\{\backslash mu\; +\; \backslash sigma^2/2\}\backslash sqrt\{e^\{\backslash sigma^2\}\; -\; 1\}$$

Inverse-variance weighted mean of measurements

$${\bar{x}}_w=\frac{\sum x_i\mathrm{/}{u}_i^2}{\sum 1\mathrm{/}{u}_i^2},u({\bar{x}}_w)=(\sum 1\mathrm{/}{u}_i^2{)}^{-1\mathrm{/}2}\backslash bar\; x\_w\; =\; \backslash frac\{\backslash sum\; x\_i/u\_i^2\}\{\backslash sum\; 1/u\_i^2\},\backslash quad\; u(\backslash bar\; x\_w)\; =\; \backslash Big(\backslash sum\; 1/u\_i^2\backslash Big)^\{-1/2\}$$

Type B standard uncertainty from a stated half-width

$$u=\frac{a}{\sqrt{3}}(\text{rect.}),\frac{a}{\sqrt{6}}(\text{tri.}),\frac{a}{\sqrt{2}}(\text{arcsine})u\; =\; \backslash frac\{a\}\{\backslash sqrt\; 3\}\backslash \; (\backslash text\{rect.\}),\backslash \; \backslash frac\{a\}\{\backslash sqrt\; 6\}\backslash \; (\backslash text\{tri.\}),\backslash \; \backslash frac\{a\}\{\backslash sqrt\; 2\}\backslash \; (\backslash text\{arcsine\})$$

Uncertainty of a function of one variable

$$u_f=\mid f^{\mathrm{\prime }}(x)\mid u_{x}u\_f\; =\; \backslash left\backslash vert\; f\text{'}(x)\backslash right\backslash vert\; u\_x$$

Uncertainty of a linear combination of correlated inputs

$$u_y^2=\sum _i\sum _j{c}_i{c}_j{u}_i{u}_j{r}_{ij}u\_y^2\; =\; \backslash sum\_i\backslash sum\_j\; c\_i\; c\_j\; u\_i\; u\_j\; r\_\{ij\}$$

Uncertainty of a power x^n

$$\frac{u_y}{\mathrm{\mid }y\mathrm{\mid }}=\mathrm{\mid }n\mathrm{\mid }\frac{u_x}{\mathrm{\mid }x\mathrm{\mid }}\backslash frac\{u\_y\}\{\backslash vert\; y\backslash vert\; \}\; =\; \backslash vert\; n\backslash vert\; \backslash ,\backslash frac\{u\_x\}\{\backslash vert\; x\backslash vert\; \}$$

Uncertainty of a power-law product y = k x1^p1 x2^p2 ...

$$\frac{u_y}{\mathrm{\mid }y\mathrm{\mid }}=\sqrt{\sum _i{\left(p_i\frac{u_i}{x_i}\right)}^2}\backslash frac\{u\_y\}\{\backslash vert\; y\backslash vert\; \}\; =\; \backslash sqrt\{\backslash sum\_i\; \backslash left(p\_i\backslash ,\backslash frac\{u\_i\}\{x\_i\}\backslash right)^2\}$$

Uncertainty of a product and a quotient

$$\frac{u_{ab}}{\mathrm{\mid }ab\mathrm{\mid }}=\frac{u_{a\mathrm{/}b}}{\mathrm{\mid }a\mathrm{/}b\mathrm{\mid }}=\sqrt{{\left(\frac{u_a}{a}\right)}^2+{\left(\frac{u_b}{b}\right)}^2}\backslash frac\{u\_\{ab\}\}\{\backslash vert\; ab\backslash vert\; \}\; =\; \backslash frac\{u\_\{a/b\}\}\{\backslash vert\; a/b\backslash vert\; \}\; =\; \backslash sqrt\{\backslash left(\backslash frac\{u\_a\}\{a\}\backslash right)^2\; +\; \backslash left(\backslash frac\{u\_b\}\{b\}\backslash right)^2\}$$

Uncertainty of a product and quotient with correlation

$$u_{ab}^2=b^2{u}_a^2+a^2{u}_b^2+2ab\rho u_a{u}_{b}u\_\{ab\}^2\; =\; b^2u\_a^2\; +\; a^2u\_b^2\; +\; 2ab\backslash ,\backslash rho\backslash ,u\_au\_b$$

Uncertainty of a sum or linear combination (independent inputs)

$$u_y=\sqrt{\sum _i{c}_i^2{u}_i^2}u\_y\; =\; \backslash sqrt\{\backslash sum\_i\; c\_i^2\; u\_i^2\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified bayesian statistics capabilities, and any question whose stated numbers it cannot all use: it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (Bayes' theorem, conjugate posteriors, credible intervals, Bayes factors, decision analysis, uncertainty propagation, Monte Carlo and sensitivity analysis); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused; use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{bayesian_statistics_ai999x_1_5_0,
  title  = {bayesian-statistics-ai999x: an offline, verified bayesian statistics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.5.0},
  url    = {https://huggingface.co/AI999X/bayesian-statistics-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/bayesian-statistics-ai999x/discussions or visit https://ai999x.com.

Support the project

bayesian-statistics-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com