Monte Carlo Estimation of Bayesian Credible and HPD Intervals

TLDR

The study develops Monte Carlo techniques to estimate Bayesian credible and highest‑probability‑density intervals for parameters of interest. These techniques use Markov chain Monte Carlo samples and importance‑sampling draws to approximate the posterior intervals, and are applied to a Bayesian hierarchical model with analytically intractable integrals. The methods successfully compute HPD intervals for both parameters and functions of parameters, as demonstrated by theoretical development and simulation examples.

Abstract

Abstract This article considers how to estimate Bayesian credible and highest probability density (HPD) intervals for parameters of interest and provides a simple Monte Carlo approach to approximate these Bayesian intervals when a sample of the relevant parameters can be generated from their respective marginal posterior distribution using a Markov chain Monte Carlo (MCMC) sampling algorithm. We also develop a Monte Carlo method to compute HPD intervals for the parameters of interest from the desired posterior distribution using a sample from an importance sampling distribution. We apply our methodology to a Bayesian hierarchical model that has a posterior density containing analytically intractable integrals that depend on the (hyper) parameters. We further show that our methods are useful not only for calculating the HPD intervals for the parameters of interest but also for computing the HPD intervals for functions of the parameters. Necessary theory is developed and illustrative examples—including a simulation study—are given.