UC Riverside

Markov chain Monte Carlo (MCMC) is a sampling technique that allows for estimating features of intractable probability distributions. Output analysis of MCMC samples aims to assess the quality of the sampler and the resulting estimates. We provide an example based overview of current best practices using visualizations and the estimation of features of interest using Markov chain central limit theorems. Most features of interest are functionals of expectations or quantiles. The estimation of quantiles has thus-far been limited to univariate theory and marginal limiting distributions, while the estimation of expectations enjoys multivariate approaches through a joint limiting distribution. In this work, we provide an extension to jointly estimate combinations of functionals of expectations and quantiles through a joint limiting distribution for the Monte Carlo error. We use this limiting distribution to establish a procedure for finding sets of simultaneous intervals forming confidence regions of approximately correct coverage probabilities. These simultaneous intervals motivate a class of visualizations for Monte Carlo errors for a broad class of estimation procedures for which a multivariate normal limiting distribution holds. Finally, we consider MCMC sample size through sequential stopping rules which terminate simulation once the Monte Carlo errors become suitably small. We develop a general sequential stopping rule for combinations of expectations and quantiles from Markov chain output and provide a simulation study to illustrate the validity.