  We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte
Carlo methods, such as the Metropolis algorithm. The problem is to compute the
expectation (or integral) of f with respect to a measure which can be given by
a density with respect to another measure. A straight simulation of the desired
distribution by a random number generator is in general not possible. Thus it
is reasonable to use Markov chain sampling with a burn-in. We study such an
algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an
explicit error bound.
