Abstract
We derive explicit asymptotic confidence intervals for any Markov chain Monte Carlo (MCMC) algorithm with finite asymptotic variance, started at any initial state, without requiring a Central Limit Theorem nor reversibility nor geometric ergodicity nor any bias bound. We also derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity. We relate those non-asymptotic bounds to properties of MCMC bias, and show that polynomially ergodicity implies certain bias bounds. We also apply our results to several numerical examples. It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when CLTs are not available.
Highlights
Markov chain Monte Carlo (MCMC) is a very powerful tool for estimating and sampling from complicated high-dimensional distributions (see e.g. (Brooks, Gelman, Jones, & Meng, 2011) and the many references therein)
We derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity
It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when Central Limit Theorem (CLT) are not available
Summary
Markov chain Monte Carlo (MCMC) is a very powerful tool for estimating and sampling from complicated high-dimensional distributions (see e.g. (Brooks, Gelman, Jones, & Meng, 2011) and the many references therein). The majority of the existing results for quantifying MCMC accuracy rely heavily on the Markov chain Central Limit Theorem (CLT) This CLT is only known to be valid under specific conditions like geometric ergodicity or reversibility, which do not always hold and can be difficult to verify Afterwards, the regeneration epochs are viewed as independent random variables in order to derive the confidence interval All of these methods require assumptions or calculations which can make them difficult to implement in practice.
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