Perturbation theory for Markov chains addresses the question of how small differences in the transition probabilities of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the nth 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.

We prove explicit, i.e. non-asymptotic, error bounds for Markov chain Monte Carlo methods. The problem is to compute the expectation of a function f with respect to a measure π. Different convergence properties of Markov chains imply different error bounds. For uniformly ergodic and reversible Markov chains we prove a lower and an upper error bound with respect to f 2 . If there exists an L2-spectral gap, which is a weaker convergence property than uniform ergodicity, then we show an upper error bound with respect to f p for p > 2. Usually a burn-in period is an efficient way to tune the algorithm. We provide and justify a recipe how to choose the burn-in period. The error bounds are applied to the problem of the integration with respect to a possibly unnormalized density. More precise, we consider the integration with respect to logconcave densities and the integration over convex bodies. By the use of the Metropolis algorithm based on a ball walk and the hit-and-run algorithm it is shown that both problems are polynomial tractable.

The problem of finding the largest empty axis-parallel box amidst a point configuration is a classical problem in computational geometry. It is known that the volume of the largest empty box is of asymptotic order 1/n for n → ∞ and fixed dimension d. However, it is natural to assume that the volume of the largest empty box increases as d gets larger. In the present paper we prove that this actually is the case: for every set of n points in [0, 1] d there exists an empty box of volume at least c d n −1 , where c d → ∞ as d → ∞. More precisely, c d is at least of order roughly log d.

Metropolis algorithms for approximate sampling of probability measures on infinite dimensional Hilbert spaces are considered and a generalization of the preconditioned Crank-Nicolson (pCN) proposal is introduced. The new proposal is able to incorporate information on the measure of interest. A numerical simulation of a Bayesian inverse problem indicates that a Metropolis algorithm with such a proposal performs independently of the state space dimension and the variance of the observational noise. Moreover, a qualitative convergence result is provided by a comparison argument for spectral gaps. In particular, it is shown that the generalization inherits geometric convergence from the Metropolis algorithm with pCN proposal.

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