A Guide to Exact Simulation

Abstract

Markov Chain Monte Carlo (MCMC) methods are used to sample from complicated multivariate distributions with normalizing constants that may not be computable in practice and from which direet sampling is not feasible. A fundamental problem is to determine convergence of the chains. Propp & Wilson (1996) devised a Markov chain algorithm called Coupling From The Past (CFTP) that solves this problem, as it produces exact samples from the target distribution and determines automatidy how long it needs to run. Exact sampling by CF"P and other methods is currently a thriving resuuch topic. This paper gives a review of some of thee ideas, with emphasis on the CFI'P dgorith. The concepts of coupling and monotone CFTP are introduced, and results on the running time of the algorithm presented. The interruptible method of F i l l (1998) and the method of Murdoch & Green (1998) for exact sampling for continuous distributions are presented. Novel simulation experiments are reported for exact sampling from the Ising model in the setting of Bayesian image restoration, and the results are compared to standard MCMC. The results show that C F " worka at least as well as standard MCMC, with convergence monitored by the method d M e r y & Lewis (19!E, 199a).

Probability for Statistics and Machine Learning

“…Diaconis et al (2008) is a tour de force on convergence of the Gibbs sampler, and Diaconis and Saloff-Coste (1998) on the Metropolis algorithm. Other important references on the difficult issue of convergence are Diaconis and Stroock (1991), Tierney (1994), Athreya, Doss and Sethuraman (1996), Propp andWilson (1998), andRosenthal (2002); Dimakos (2001) is a useful survey. Various useful modifications of the basic MCMC have been suggested to address specific important applications.…”

The Complexity of Ferromagnetic Ising with Local Fields

“…The most commonly studied version of the Ising model (see, for example, [2]) is the version in which the interaction energies J i,j are uniform over the edges (i, j) ∈ E and the local magnetic fields v are uniform over vertices v ∈ V . The general model is also studied, particularly in the situation in which the interaction energies J i,j and the local fields v are random variables (see [1,6,14]).…”

A Short History of Markov Chain Monte Carlo: Subjective Recollections from Incomplete Data

“…Introduced in the seminal paper of Propp and Wilson (1996), perfect sampling, namely, the ability to use MCMC methods to produce an exact (or perfect) simulation from the target, maintains a unique place in the history of MCMC methods. Although this exciting discovery led to an outburst of papers, in particular, in the large body of work of Møller and coauthors, including the book by Møller and Waagepetersen (2003), as well as many reviews and introductory materials, like Casella, Lavine and Robert (2001), Fismen (1998) and Dimakos (2001), the excitement quickly dried out. The major reason for this ephemeral lifespan is that the construction of perfect samplers is most often close to impossible or impractical, despite some advances in the implementation (Fill, 1998a(Fill, , 1998b.…”