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, @math . 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 johndrow2015approximations from the uniformly ergodic case to the geometrically ergodic case. We then apply our results to the class of Noisy MCMC' algorithms.
Lastly, Rudolf and Schweizer @cite_6 also use the Wasserstein topology. They focus on approximate MCMC algorithms, with applications to autoregressive processes and stochastic Langevin algorithms for Gibbs random fields. Their results use the following assumptions: the original kernel is Wasserstein ergodic; a Lyapunov drift condition for perturbed kernel is given, with drift function @math ; @math has finite expectation under the initial distribution; and the perturbation operator is uniformly bounded in a @math -normalised Wasserstein norm. Their main results are: explicit bounds on the Wasserstein distance between the original and perturbed finite time sampling distributions; and explicit bounds on the Wasserstein distance between stationary distributions.
@cite_6
2120918452
Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the @math th step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings and stochastic Langevin algorithms.