Mixing time estimation in reversible Markov chains from a single sample path

Daniel Hsu, Aryeh Kontorovich, Csaba Szepesvári

Research output: Contribution to journalConference articlepeer-review

22 Scopus citations

Abstract

This article provides the first procedure for computing a fully data-dependent interval that traps the mixing time tmix of a finite reversible ergodic Markov chain at a prescribed confidence level. The interval is computed from a single finite-length sample path from the Markov chain, and does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time trelax, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a √n rate, where n is the length of the sample path. Upper and lower bounds are given on the number of samples required to achieve constant-factor multiplicative accuracy. The lower bounds indicate that, unless further restrictions are placed on the chain, no procedure can achieve this accuracy level before seeing each state at least Ω(trelax) times on the average. Finally, future directions of research are identified.

Original languageEnglish
Pages (from-to)1459-1467
Number of pages9
JournalAdvances in Neural Information Processing Systems
Volume2015-January
StatePublished - 1 Jan 2015
Event29th Annual Conference on Neural Information Processing Systems, NIPS 2015 - Montreal, Canada
Duration: 7 Dec 201512 Dec 2015

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