Error Bounds for Approximations of Geometrically Ergodic Markov Chains

Jeffrey Negrea, Jeffrey S. Rosenthal

Published 2017-02-24Version 1

A common tool in the practice of Markov Chain Monte Carlo is to use approximating transition kernels to speed up computation when the true kernel is slow to evaluate. A relatively limited set of quantitative tools exist to determine whether the performance of such approximations will be well behaved and to assess the quality of approximation. We derive a set a tools for such analysis based on the Hilbert space generated by the stationary distribution we intend to sample, $L_2(\pi)$. The focus of our work is on determining whether the approximating kernel (i.e.\ perturbation) will preserve the geometric ergodicity of the chain, and whether the approximating stationary distribution will be close to the original stationary distribution. Our results directly generalise the results of \cite{johndrow2015approximations} from the uniformly ergodic case to the geometrically ergodic case. We then apply our results to the class of `Noisy MCMC' algorithms.