• Bernoulli
  • Volume 22, Number 3 (2016), 1808-1838.

Markov Chain Monte Carlo confidence intervals

Yves F. Atchadé

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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.

Article information

Bernoulli, Volume 22, Number 3 (2016), 1808-1838.

Received: February 2014
Revised: October 2014
First available in Project Euclid: 16 March 2016

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Zentralblatt MATH identifier

Berry–Esseen bounds confidence interval lag-window estimators martingale approximation MCMC reversible Markov chains


Atchadé, Yves F. Markov Chain Monte Carlo confidence intervals. Bernoulli 22 (2016), no. 3, 1808--1838. doi:10.3150/15-BEJ712.

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