Approximations of Geometrically Ergodic Reversible Markov Chains

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.

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• The pseudo-marginal approach for efficient Monte Carlo computations
• Renewal theory and computable convergence rates for geometrically ergodic Markov chains
• Automated Scalable Bayesian Inference via Hilbert Coresets
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• Pólya Urn Latent Dirichlet Allocation: a doubly sparse massively parallel sampler
• Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance
• Convergence Speed and Approximation Accuracy of Numerical MCMC
• Perturbation theory for killed Markov processes and quasi-stationary distributions