Markov Chain Monte Carlo confidence intervals

Abstract

For a reversible and ergodic Markov chain $\{X_n,n\ge 0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi (h)$ can be constructed whenever the asymptotic variance ${\sigma }_P^2(h)$ 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.

Published at http://dx.doi.org/10.3150/15-BEJ712 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Batch means and spectral variance estimators in Markov chain Monte Carlo
• On mean central limit theorems for stationary sequences
• Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo