Bayesian

Overview

Bayesian inference provides a principled framework for updating uncertainty as new evidence arrives. Here, Bayesian inference tools turn prior assumptions and observed data into posterior distributions, interval estimates, and summaries. These methods matter in analytics because they produce interpretable probability statements for risk, reliability, and forecasting.

Core Concepts: The shared structure is posterior updating, where p(\theta\mid x) \propto p(x\mid\theta)\,p(\theta). Practical workflows emphasize conjugate priors (closed-form updates), credible intervals (posterior probability statements), and posterior summaries (means, variances, entropy, MAP, and tail probabilities). Numerical stability is central, especially for log-domain normalization and special-function terms such as log-Beta and log-sum-exp.

Implementation: These tools are primarily implemented with SciPy, especially scipy.stats and scipy.special, with selected quantile utilities from NumPy. SciPy supplies robust distribution objects and special functions, while NumPy supports efficient array handling and empirical posterior summaries.

Conjugate Priors: The conjugate-update tools cover Beta-Binomial and Normal/Gamma-family workflows where posterior parameters remain in the same family as the prior. BB_POST_UPDATE and BB_QBETA update binomial success models and extract posterior quantiles, while BB_LOGBETA supplies stable normalization terms for evidence and marginal-likelihood calculations. For positive-scale parameters, GAMMA_POST_Q and INVGAMMA_POST_Q provide posterior quantiles under Gamma and inverse-Gamma assumptions. Normal-model updates are handled by NIG_POST_UPDATE and NN_POST_UPDATE, common in process monitoring and Bayesian calibration.

Credible Intervals: This group focuses on posterior interval construction across common model families and sample-based inference. BAYES_MVS_CI and MVSDIST_CI provide Bayesian intervals for mean, variance, and standard deviation from observed data under SciPy’s Bayesian summary distributions. Parameter-specific interval tools include BETA_CI_BOUNDS, GAMMA_CI_BOUNDS, and INVGAMMA_CI_BOUNDS, which translate posterior hyperparameters into equal-tailed bounds for proportions, rates, and scales. For simulation-based pipelines, SAMPLE_EQTAIL_CI and SAMPLE_HPD_CI summarize posterior draws into equal-tailed and approximate HPD intervals used in Bayesian Monte Carlo reporting.

Dirichlet Multinomial: These functions support categorical Bayesian modeling where counts update simplex-valued probability vectors. DM_POST_UPDATE performs Dirichlet posterior updating, DM_PREDICTIVE returns posterior predictive category probabilities, and DM_CRED_INT reports category-wise credible intervals. Distribution diagnostics and normalization helpers are provided by DM_DIRICHLET_SUM, DM_LOGBETA, and DM_LOGSUM_NORM. These are useful for topic proportions, survey share estimation, and multinomial forecasting.

Posterior Summarization: After obtaining posterior values, this group converts them into stable decision metrics and information-theoretic diagnostics. POSTERIOR_BMV and POSTERIOR_WMEANVAR summarize central tendency and dispersion, while POSTERIOR_MAP and POSTERIOR_TAILPROB support threshold-based decisions and risk checks. Information and log-domain utilities are covered by POSTERIOR_ENTROPY, POSTERIOR_LOGSUMEXP, and POSTERIOR_XLOGY, essential for stable normalization, evidence calculations, and posterior table diagnostics. These summaries are typically the final layer in dashboards and model comparison workflows.

Conjugate Priors

Tool	Description
BB_LOGBETA	Compute the log-Beta term used in conjugate posterior calculations.
BB_POST_UPDATE	Update Beta-Binomial posterior hyperparameters from observed counts.
BB_QBETA	Compute a Beta posterior quantile for Beta-Binomial models.
GAMMA_POST_Q	Compute a Gamma posterior quantile from shape-rate parameters.
INVGAMMA_POST_Q	Compute an inverse-Gamma posterior quantile.
NIG_POST_UPDATE	Update Normal-Inverse-Gamma posterior hyperparameters from sample summaries.
NN_POST_UPDATE	Update Normal posterior parameters for unknown mean with known variance.

Credible Intervals

Tool	Description
BAYES_MVS_CI	Compute Bayesian credible intervals for mean, variance, and standard deviation from sample data.
BETA_CI_BOUNDS	Compute an equal-tailed Bayesian credible interval for a proportion using a Beta posterior.
GAMMA_CI_BOUNDS	Compute an equal-tailed Bayesian credible interval for a positive rate parameter using Gamma quantiles.
INVGAMMA_CI_BOUNDS	Compute an equal-tailed Bayesian credible interval for a positive scale or variance parameter using Inverse-Gamma quantiles.
MVSDIST_CI	Compute Bayesian credible intervals from posterior distributions of mean, variance, and standard deviation.
SAMPLE_EQTAIL_CI	Compute an equal-tailed credible interval from posterior samples using empirical quantiles.
SAMPLE_HPD_CI	Approximate a highest posterior density interval from posterior samples using the narrowest empirical window.

Dirichlet Multinomial

Tool	Description
DM_CRED_INT	Compute category-wise credible intervals from posterior Dirichlet parameters.
DM_DIRICHLET_SUM	Compute Dirichlet density and moments for a category-probability vector.
DM_LOGBETA	Compute the Dirichlet log-normalization term using log-gamma values.
DM_LOGSUM_NORM	Compute a stable log normalizer and normalized probabilities from log-values.
DM_POST_UPDATE	Update Dirichlet posterior parameters from prior hyperparameters and observed counts.
DM_PREDICTIVE	Compute posterior predictive category probabilities from Dirichlet parameters.

Posterior Summarization

Tool	Description
POSTERIOR_BMV	Compute Bayesian posterior summaries for mean, variance, and standard deviation.
POSTERIOR_ENTROPY	Compute Shannon or relative entropy for posterior probability tables.
POSTERIOR_LOGSUMEXP	Compute stable log-sum-exp aggregates for posterior normalization and evidence calculations.
POSTERIOR_MAP	Extract the MAP estimate from a tabulated posterior distribution.
POSTERIOR_TAILPROB	Compute posterior tail probabilities relative to a decision threshold.
POSTERIOR_WMEANVAR	Compute posterior weighted mean and variance summaries from values and weights.
POSTERIOR_XLOGY	Compute numerically stable x times log y terms for posterior information calculations.