'mcref.bib'
title: Estimating Monte Carlo variance from multiple Markov chains
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Introduction
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Markov chain Monte Carlo (MCMC) algorithms have emerged as an essential tool for parameter estimation of target distributions when obtaining independent samples is inefficient. With increased computational power and the presence of multiple cores on personal computers, running parallel Markov chains dispersed over the state space is common for improved inference. On the other hand, output analysis in MCMC has predominantly focused on single-chain MCMC output with the understanding that multiple-chain analysis can follow from a simple averaging of information. We demonstrate that a simple averaging of the Monte Carlo limiting covariance matrix can yield significant underestimation, warranting specific methods for multiple-chain output analysis.

Let $F$ be the target distribution defined on a $d$-dimensional space ${\mathcal{X}}$, equipped with a countably generated $\sigma$-field. Let function $h:{\mathcal{X}}\to \mathbb{R}^p$ be such that $\mu = \int_{{\mathcal{X}}} h(x) F(dx)$ is a quantity of interest. Let $\{X_{1,t}\}_{t\geq 1}, \dots, \{X_{m,t}\}_{t\geq 1}$ be $m$ independent $F$-ergodic Markov chains. The Monte Carlo estimator of $\mu$ from the $k$th Markov chain is $$\hat{\mu}_{k} = \dfrac{1}{n} {\displaystyle}\sum_{t=1}^{n} h\left(X_{k,t} \right) \overset{a.s.}{\to} \mu \text{ as } n \to \infty\,,$$ where the convergence holds for any starting value. Consequently, the 