Approximations of Geometrically Ergodic Reversible Markov Chains

TL;DR

This paper develops analytical tools using Hilbert space methods to evaluate the accuracy and efficiency of approximate Markov Chain Monte Carlo kernels, focusing on preserving geometric ergodicity and stationarity.

Contribution

It extends existing results from uniformly ergodic to geometrically ergodic reversible chains, providing new criteria for approximation quality in MCMC.

Findings

Approximate kernels can preserve geometric ergodicity under certain conditions.

The stationary distribution of the approximation can be close to the true distribution.

Tools are applicable to various approximate MCMC algorithms.

Abstract

A common tool in the practice of Markov Chain Monte Carlo is to use approximating transition kernels to speed up computation when the desired kernel is slow to evaluate or intractable. A limited set of quantitative tools exist to assess the relative accuracy and efficiency of such approximations. We derive a set of tools for such analysis based on the Hilbert space generated by the stationary distribution we intend to sample, $L_2(\pi)$. Our results apply to approximations of reversible chains which are geometrically ergodic, as is typically the case for applications to Markov Chain Monte Carlo. The focus of our work is on determining whether the approximating kernel will preserve the geometric ergodicity of the exact chain, and whether the approximating stationary distribution will be close to the original stationary distribution. For reversible chains, our results extend the results of Johndrow et al. [18] from the uniformly ergodic case to the geometrically ergodic case, under some additional regularity conditions. We then apply our results to a number of approximate MCMC algorithms.