Open Access
Translator Disclaimer
February 1998 An interruptible algorithm for perfect sampling via Markov chains
James Allen Fill
Ann. Appl. Probab. 8(1): 131-162 (February 1998). DOI: 10.1214/aoap/1027961037

Abstract

For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution $\pi$ on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate normalizing constant. The same difficulty arises in computer science problems where one seeks to sample randomly from a large finite distributive lattice whose precise size cannot be ascertained in any reasonable amount of time.

The Markov chain Monte Carlo (MCMC) approximate sampling approach to such a problem is to construct and run "for a long time" a Markov chain with long-run distribution $\pi$. But determining how long is long enough to get a good approximation can be both analytically and empirically difficult.

Recently, Propp and Wilson have devised an ingenious and efficient algorithm to use the same Markov chains to produce perfect (i.e., exact) samples from $\pi$. However, the running time of their algorithm is an unbounded random variable whose order of magnitude is typically unknown a priori and which is not independent of the state sampled, so a naive user with limited patience who aborts a long run of the algorithm will introduce bias.

We present a new algorithm which (1) again uses the same Markov chains to produce perfect samples from $\pi$, but is based on a different idea (namely, acceptance/rejection sampling); and (2) eliminates user-impatience bias. Like the Propp-Wilson algorithm, the new algorithm applies to a general class of suitably monotone chains, and also (with modification) to "anti-monotone" chains. When the chain is reversible, naive implementation of the algorithm uses fewer transitions but more space than Propp-Wilson. When fine-tuned and applied with the aid of a typical pseudorandom number generator to an attractive spin system on n sites using a random site updating Gibbs sampler whose mixing time $\tau$ is polynomial in n, the algorithm runs in time of the same order (bound) as Propp-Wilson [expectation $O(\tau \log n)$] and uses only logarithmically more space [expectation $O(n \log n)$, vs.$O9n)$ for Propp-Wilson].

Citation

Download Citation

James Allen Fill. "An interruptible algorithm for perfect sampling via Markov chains." Ann. Appl. Probab. 8 (1) 131 - 162, February 1998. https://doi.org/10.1214/aoap/1027961037

Information

Published: February 1998
First available in Project Euclid: 29 July 2002

zbMATH: 0939.60084
MathSciNet: MR1620346
Digital Object Identifier: 10.1214/aoap/1027961037

Subjects:
Primary: 60J10
Secondary: 60G40 , 62D05 , 68U20

Keywords: attractive spin system , Duality , Gibbs sampler , Ising model , Markov chain Monte Carlo , monotone chain , ‎partially ordered set , perfect simulation , rejection sampling , separation , strong stationary time

Rights: Copyright © 1998 Institute of Mathematical Statistics

JOURNAL ARTICLE
32 PAGES


SHARE
Vol.8 • No. 1 • February 1998
Back to Top