## Abstract

This article provides the first procedure for computing a fully data-dependent interval that traps the mixing time t_{mix} 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 t_{relax}, 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(t_{relax}) times on the average. Finally, future directions of research are identified.

Original language | English |
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Pages (from-to) | 1459-1467 |

Number of pages | 9 |

Journal | Advances in Neural Information Processing Systems |

Volume | 2015-January |

State | Published - 1 Jan 2015 |

Event | 29th Annual Conference on Neural Information Processing Systems, NIPS 2015 - Montreal, Canada Duration: 7 Dec 2015 → 12 Dec 2015 |