Markov chain Monte Carlo confidence intervals (Q282567)

Given a Markov chain \(\{X_n: n \geq 0\}\) with invariant distribution \(\pi\), denote the asymptotic expectation by \(\pi(h)=\int h(z) \pi(dz)\) and the asymptotic variance of \(h\) by \(\sigma_p^2(h)\). The classical confidence interval procedure for \(\pi(h)\) based on a consistent estimation of \(\sigma_p(h)\) requires assumptions on the convergence rate of the Markov chain, typically it is geometric ergodicity. In this article, it is shown that for a reversible Markov chain the confidence interval for \(\pi(h)\) can be constructed whenever \(0<\sigma_p(h)<\infty\) without any additional requirements on the convergence rate. Instead of consistency, the confidence interval is derived using the so-called fixed-b lag-window estimation of \(\sigma_p^2(h)\). It is also shown that the convergence rate of the proposed procedure is faster than the convergence rate of the classical confidence interval procedure.