Explicit error bounds for Markov chain Monte Carlo
This work offers rigorous error guarantees for MCMC practitioners, but the results are incremental as they extend known asymptotic bounds to explicit forms.
The paper provides explicit, non-asymptotic error bounds for Markov chain Monte Carlo methods, showing that integration with respect to log-concave densities and over convex bodies is polynomially tractable using specific 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 the L2 -norm of f . 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 the Lp -norm of f 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 log-concave 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.