Explicit error bounds for Markov chain Monte Carlo

TL;DR

This paper establishes explicit, non-asymptotic error bounds for Markov chain Monte Carlo methods, providing practical guidelines for choosing burn-in periods and demonstrating polynomial tractability for specific integration problems.

Contribution

It derives explicit error bounds for MCMC under various convergence conditions and offers a practical approach for burn-in selection, with applications to log-concave densities and convex bodies.

Findings

Error bounds for uniformly ergodic and reversible chains with L2-norm

Error bounds under spectral gap condition with Lp-norm

Polynomial tractability for integration over log-concave densities and convex bodies

Abstract

We prove explicit, i.e. non-asymptotic, error bounds for Markov chain Monte Carlo methods. The problem is to compute the expectation of a function f with respect to a measure {\pi}. Different convergence properties of Markov chains imply different error bounds. For uniformly ergodic and reversible Markov chains we prove a lower and an upper error bound with respect to the L2 -norm of f . If there exists an L2 -spectral gap, which is a weaker convergence property than uniform ergodicity, then we show an upper error bound with respect to the Lp -norm of f for p > 2. Usually a burn-in period is an efficient way to tune the algorithm. We provide and justify a recipe how to choose the burn-in period. The error bounds are applied to the problem of the integration with respect to a possibly unnormalized density. More precise, we consider the integration with respect to log-concave densities and the integration over convex bodies. By the use of the Metropolis algorithm based on a ball walk and the hit-and-run algorithm it is shown that both problems are polynomial tractable.