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Biometrika 2006 93(2):451-458; doi:10.1093/biomet/93.2.451
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© 2006 Biometrika Trust

Miscellanea

An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants

J. Møller1, A. N. Pettitt2, R. Reeves2 and K. K. Berthelsen3

1 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg E, Denmark. jm{at}math.aau.dk, 2 School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Queensland 4001, Australia. a.pettitt{at}qut.edu.au, r.reeves{at}qut.edu.au, 3 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg E, Denmark. kkb{at}math.aau.dk

Maximum likelihood parameter estimation and sampling from Bayesian posterior distributions are problematic when the probability density for the parameter of interest involves an intractable normalising constant which is also a function of that parameter. In this paper, an auxiliary variable method is presented which requires only that independent samples can be drawn from the unnormalised density at any particular parameter value. The proposal distribution is constructed so that the normalising constant cancels from the Metropolis-Hastings ratio. The method is illustrated by producing posterior samples for parameters of the Ising model given a particular lattice realisation.

Key Words: Auxiliary variable method; Ising model; Markov chain Monte Carlo; Metropolis-Hastings algorithm; Normalising constant; Partition function.

Received February 2004. Revised November 2005.


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