## Electronic Communications in Probability

### How to Combine Fast Heuristic Markov Chain Monte Carlo with Slow Exact Sampling

#### Abstract

Given a probability law $\pi$ on a set $S$ and a function $g : S \rightarrow R$, suppose one wants to estimate the mean $\bar{g} = \int g d\pi$. The Markov Chain Monte Carlo method consists of inventing and simulating a Markov chain with stationary distribution $\pi$. Typically one has no a priori bounds on the chain's mixing time, so even if simulations suggest rapid mixing one cannot infer rigorous confidence intervals for $\bar{g}$. But suppose there is also a separate method which (slowly) gives samples exactly from $\pi$. Using $n$ exact samples, one could immediately get a confidence interval of length $O(n^{-1/2})$. But one can do better. Use each exact sample as the initial state of a Markov chain, and run each of these $n$ chains for $m$ steps. We show how to construct confidence intervals which are always valid, and which, if the (unknown) relaxation time of the chain is sufficiently small relative to $m/n$, have length $O(n^{-1} \log n)$ with high probability.

#### Article information

Source
Electron. Commun. Probab. Volume 6 (2001), paper no. 8, 79-89.

Dates
Accepted: 28 July 2001
First available in Project Euclid: 19 April 2016

https://projecteuclid.org/euclid.ecp/1461097553

Digital Object Identifier
doi:10.1214/ECP.v6-1037

Mathematical Reviews number (MathSciNet)
MR1855344

Zentralblatt MATH identifier
0987.60084

Rights

#### Citation

Bandyopadhyay, Antar; Aldous, David. How to Combine Fast Heuristic Markov Chain Monte Carlo with Slow Exact Sampling. Electron. Commun. Probab. 6 (2001), paper no. 8, 79--89. doi:10.1214/ECP.v6-1037. https://projecteuclid.org/euclid.ecp/1461097553.

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