This summary explores the foundations and applications of Bayesian Credible Intervals (BCIs), distinguishing them from frequentist confidence intervals. BCIs are defined as ranges within which a parameter lies with a specified posterior probability, given the observed data and prior beliefs.
Unlike confidence intervals, which rely on hypothetical repeated sampling, BCIs directly reflect subjective belief about parameter location: \[ \text{Pr}(\theta \in [a, b] \mid \text{data}) = 0.95 \] where \( [a, b] \) is the 95% credible interval.
The article explains how BCIs are computed from posterior distributions, often using conjugate priors (e.g., normal-normal or beta-binomial models). Two canonical examples are discussed:
The paper introduces Highest Density Intervals (HDIs) as a refinement of BCIs in asymmetric distributions—intervals that always contain the most probable values. Graphical examples illustrate how HDIs adapt to skewed posteriors.
Additional sections compare BCIs with confidence intervals and explore their behavior under increasing sample size, showing asymptotic agreement when non-informative priors are used. Discrepancies between BCIs and Bayes Factors are also examined, particularly in light of Type I and Type II conflicts where BF and interval-based conclusions diverge.
The article closes with discussion of computational tools (e.g., MCMC, R packages like rjags, brms, BayesFactor) and recommendations for practitioners.
For full details and mathematical derivations, see the original entry in the International Encyclopedia of Statistical Science.