Nonasymptotic bounds on the estimation error of MCMC algorithms

TL;DR

This paper derives nonasymptotic, sharp bounds on the mean square error of MCMC estimators applicable to a wide range of ergodic Markov chains, with explicit bounds under certain conditions.

Contribution

It provides the first nonasymptotic bounds on MCMC estimation error valid for all ergodic Markov chains, including explicit bounds under geometric and polynomial ergodicity.

Findings

Bounds are sharp, matching the CLT asymptotic variance term.

Explicit bounds are derived for geometrically and polynomially ergodic chains.

Results include confidence estimation for MCMC estimators.

Abstract

We address the problem of upper bounding the mean square error of MCMC estimators. Our analysis is nonasymptotic. We first establish a general result valid for essentially all ergodic Markov chains encountered in Bayesian computation and a possibly unbounded target function $f$. The bound is sharp in the sense that the leading term is exactly $\sigma_{\mathrm {as}}^2(P,f)/n$, where $\sigma_{\mathrm{as}}^2(P,f)$ is the CLT asymptotic variance. Next, we proceed to specific additional assumptions and give explicit computable bounds for geometrically and polynomially ergodic Markov chains under quantitative drift conditions. As a corollary, we provide results on confidence estimation.