Estimating Bayesian credible intervals

Under a Bayesian approach to a hierarchical model, quantile or interval estimation is often used to summarize the posterior distribution of a parameter. When using an Markov Chain Monte Carlo algorithm such as the Gibbs sampler to generate a sample from the posterior (marginal) of interest, calculations are often easier when done on a per-iteration (conditional) basis. Final estimators which are taken as a combination of values across iterations are often called "Rao–Blackwellized" and result in estimators with good variance properties. Such an approach is not yet used in the calculation of credible intervals. We derive here a weighted-average estimator of the endpoints of a credible interval which mimics this Rao–Blackwellized construction. We compare it to other alternatives including a naïve average estimator, the usual order statistics estimator, and an estimator based on density estimation. We obtain theorems showing when there is convergence to the true interval and discuss Central Limit Theorems for these estimators. Simulations for two hierarchical modeling scenarios (count data and continuous data) illustrate their numerical behaviors. An animal epidemiology example is included. The proposed estimator offers the smallest standard errors of the estimators studied, sometimes by several orders of magnitude, but can have a small bias.