Error bounds for Approximations of Markov chains
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The first part of this article gives error bounds for approximations of Markov kernels under Foster-Lyapunov conditions. The basic idea is that when both the approximating kernel and the original kernel satisfy a Foster-Lyapunov condition, the long-time dynamics of the two chains -- as well as the invariant measures, when they exist -- will be close in a weighted total variation norm, provided that the approximation is sufficiently accurate. The required accuracy depends in part on the Lyapunov function, with more stable chains being more tolerant of approximation error. We are motivated by the recent growth in proposals for scaling Markov chain Monte Carlo algorithms to large datasets by defining an approximating kernel that is faster to sample from. Many of these proposals use only a small subset of the data points to construct the transition kernel, and we consider an application to this class of approximating kernel. We also consider applications to distribution approximations in Gibbs sampling. Another application in which approximating kernels are commonly used is in Metropolis algorithms for Gaussian process models common in spatial statistics and nonparametric regression. In this setting, there are typically two sources of approximation error: discretization error and approximation of Metropolis acceptance ratios. Because the approximating kernel is obtained by discretizing the state space, it is singular with respect to the exact kernel. To analyze this application, we give additional results in Wasserstein metrics in contrast to the proceeding examples which quantified the level of approximation in a total variation norm.