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.