text: 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.
  @cite_2: The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. We propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.
summary: Close to the present paper, @cite_2 derive perturbation bounds to assess the robustness of approximate MCMC algorithms. The assumptions upon which their results rely are: the original chain is uniformly contractive in the total variation norm (this implies uniform ergodicity); and the perturbation is sufficiently small (in the operator norm induced by the total variation norm). The main results of their paper are: the perturbed kernel is uniformly contractive in the total variation norm; the perturbed stationary distribution is close to the original stationary distribution in total variation; explicit bounds on the total variation distance between finite time approximate sampling distributions and the original stationary distribution; explicit bounds on total variation difference between the original stationary distribution and the mixture of finite time approximate sampling distributions; and explicit bounds on the MSE for integral approximation using approximate kernel and the true kernel. The results derived by @cite_2 are applied within the same paper to a wide variety of approximate MCMC problems including low rank approximation to Gaussian processes and subsampling approximations.