Approximations of Geometrically Ergodic Reversible Markov Chains
Jeffrey Negrea, Jeffrey S. Rosenthal
TL;DR
The paper addresses the challenge of using approximations in Markov Chain Monte Carlo without sacrificing rapid convergence to the target distribution. It develops a rigorous $L_2(oldsymbol{ ightpi})$-based perturbation framework for reversible geometrically ergodic Markov chains, extending prior uniformly ergodic results to the GE setting and deriving explicit bounds on how close the perturbed stationary distribution is to the original. Central results show that small $L_2(oldsymbol{ ightpi})$ perturbations preserve GE with rate $1-(oldsymbol{ ate}-oldsymbol{ ightepsilon})$ and yield quantitative controls on stationary-distribution closeness and Monte Carlo error, including MSE bounds for perturbed chains. The framework is then applied to several approximate MCMC algorithms, including Noisy/MH, fixed deterministic density-approximations, and MC within Metropolis, providing practical conditions under which GE is preserved and estimation errors remain bounded. This work enables principled analysis and error control for a broad class of approximate MCMC methods in scientifically and computationally challenging settings, with direct implications for efficiency and reliability of Bayesian inference on large datasets or complex models.
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, $L_2(π)$. 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.