Bayesian Statistical Inference

A Bayesian credible interval (typically 95%) is the range containing 95% of the posterior probability; directly interpretable as: 'there is a 95% probability the true parameter lies in this interval.' A frequentist confidence interval (95% CI) is a range constructed such that, if the study were repeated infinitely, 95% of constructed CIs would contain the true parameter. These are fundamentally different: credible interval is about parameter probability; confidence interval is about procedure repeatability. In practice, for standard problems, they often overlap numerically, causing confusion. Always specify which you report.

A prior is the probability distribution over parameter values before observing data, reflecting prior knowledge or assumptions. Weakly informative priors contribute minimal information (flat distributions); informative priors incorporate genuine prior knowledge (e.g., treatment effect < 20% from previous trials). Skeptical priors strongly favor null (no effect), requiring strong data evidence to shift. Choose priors by: (1) reviewing previous literature, (2) consulting domain experts, (3) specifying ranges of plausible values, (4) testing sensitivity to alternative priors. Always report priors explicitly and justify them. If conclusions change dramatically with different reasonable priors, inference is weak and more data needed.

A Bayes factor (BF) quantifies evidence for one model over another: BF₁₀ = P(data|model 1) / P(data|model 2). BF > 1 favors model 1. Interpretation: BF = 3 = 'moderate evidence for model 1'; BF = 10 = 'strong evidence'; BF > 100 = 'decisive evidence.' Unlike p-values, BF does not require predefined significance threshold. BF can favor the null (BF < 1), detecting absence of evidence. Limitation: computing BF requires specifying prior model probabilities (which model is a priori plausible); different prior assignments yield different BF values. Use BF for model comparison when both models have clear practical motivation.

Partially. Bayesian inference does not use p-values or significance thresholds, so the traditional multiple comparisons correction (Bonferroni) is unnecessary. However, when fitting many models and selecting based on posterior criteria (DIC, WAIC), you risk overfitting and model selection bias analogous to frequentist multiple testing. Best practice: specify models a priori based on theory, not data exploration. If exploratory, apply cross-validation to assess generalization. Bayesian model averaging (combining predictions from multiple models weighted by posterior probabilities) is an elegant solution to model uncertainty.