Fixed precision MCMC estimation by median of products of averages

Abstract

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 = µ₁ ⋯ 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 θ̂, i.e. an estimator which is a `median of products of averages' (MPA). Sufficient conditions are given for θ̂ to have fixed relative precision at a given level of confidence, that is, to satisfy P(|θ̂ - θ| ≤ θ ∈) ≥ 1 - β. Our main tool is a new bound on the mean-square error, valid also for nonreversible Markov chains on a finite state space.