Fixed Precision MCMC Estimation by Median of Products of Averages

The standard Markov chain Monte Carlo method of estimating an expected value is to generate a Markov chain which converges to the target distribution and then compute correlated sample averages. In many applications the quantity of interestθis represented as a product of expected values,θ=µ1⋯µk, and a natural estimator is a product of averages. To increase the confidence level, we can compute a median of independent runs. The goal of this paper is to analyze such an estimator, i.e. an estimator which is a ‘median of products of averages’ (MPA). Sufficient conditions are given forto have fixed relative precision at a given level of confidence, that is, to satisfy. Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.

Fixed Precision MCMC Estimation by Median of Products of Averages

The standard Markov chain Monte Carlo method of estimating an expected value is to generate a Markov chain which converges to the target distribution and then compute correlated sample averages. In many applications the quantity of interest θ is represented as a product of expected values, θ = µ 1 ⋯ µ k , and a natural estimator is a product of averages. To increase the confidence level, we can compute a median of independent runs. The goal of this paper is to analyze such an estimator , i.e. an estimator which is a ‘median of products of averages’ (MPA). Sufficient conditions are given for to have fixed relative precision at a given level of confidence, that is, to satisfy . Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.