Markov Chain Monte Carlo confidence intervals

TL;DR

This paper develops a method for constructing valid confidence intervals for expectations under a Markov chain's invariant distribution, using a fixed-b lag-window estimator, without requiring specific convergence rates.

Contribution

It introduces a new confidence interval construction for Markov chain expectations that does not depend on the chain's convergence rate, improving over classical methods.

Findings

Confidence intervals are valid when asymptotic variance is finite and positive.

The proposed method converges faster than classical Gaussian-based confidence intervals.

No additional convergence rate conditions are required for the Markov chain.

Abstract

For a reversible and ergodic Markov chain $\{X_n,n\geq0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi(h)$ can be constructed whenever the asymptotic variance $\sigma^2_P(h)$ is finite and positive. We do not impose any additional condition on the convergence rate of the Markov chain. The confidence interval is derived using the so-called fixed-b lag-window estimator of $\sigma_P^2(h)$. We also derive a result that suggests that the proposed confidence interval procedure converges faster than classical confidence interval procedures based on the Gaussian distribution and standard central limit theorems for Markov chains.