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 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 integration with respect to a possibly unnormalized density. More precise, we consider integration with respect to log-concave densities and 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.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm485-0-1,
author = {Daniel Rudolf},
title = {Explicit error bounds for Markov chain Monte Carlo},
series = {GDML\_Books},
year = {2012},
zbl = {1273.60090},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm485-0-1}
}
Daniel Rudolf. Explicit error bounds for Markov chain Monte Carlo. GDML_Books (2012), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm485-0-1/