Monte Carlo Standard Errors for Markov Chain Monte Carlo

Markov chain Monte Carlo (MCMC) is a method of producing a correlated sample to estimate characteristics of a target distribution. A fundamental question is how long should the simulation be run? One method to address this issue is to run the simulation until the width of a confidence interval for the quantity of interest is below a user-specified value. The use of this fixed-width methods requires an estimate of the Monte Carlo standard error (MCSE). This dissertation begins by discussing why MCSEs are important, how they can be easily calculated in MCMC and how they can be used to decide when to stop the simulation. The use of MCSEs is then compared to a popular alternative in the context of multiple examples. This dissertation continues by discussing the relevant Markov chain theory with particular attention paid to the conditions and definitions needed to establish a Markov chain central limit theorem. Estimating MCSEs requires estimating the associated asymptotic variance. I introduce several techniques for estimating MCSEs: batch means, overlapping batch means, regeneration, subsampling and spectral variance estimation. Asymptotic properties useful in MCMC settings are established for these variance estimators. Specifically, I established conditions under which the estimator for the asymptotic variance in a Markov chain central limit theorem is strongly consistent. Strong consistency ensures that confidence intervals formed will be asymptotically valid. In addition, I established conditions to ensure mean-square consistency for the estimators using batch means and overlapping batch means. Mean-square coniii iv sistency is useful in choosing an optimal batch size for MCMC practitioners. Several approaches have been introduced dealing with the special case of estimating ergodic averages and their corresponding standard errors. Surprisingly, very little attention has been given to characteristics of the target distribution that cannot be represented as ergodic averages. To this end, I proposed use of subsampling methods as a means for estimating the qth quantile of the posterior distribution. Finally, the finite sample properties of subsampling are examined.