# MCMC for doubly-intractable distributions

Iain Murray, Zoubin Ghahramani, David J.C. MacKay.

Markov Chain Monte Carlo (MCMC) algorithms are routinely used to draw samples from distributions with intractable normalization constants. However, standard MCMC algorithms do not apply to doubly-intractable distributions in which there are additional parameter-dependent normalization terms; for example, the posterior over parameters of an undirected graphical model. An ingenious auxiliary-variable scheme (Møller et al., 2004) offers a solution: exact sampling (Propp and Wilson, 1996) is used to sample from a Metropolis–Hastings proposal for which the acceptance probability is tractable. Unfortunately the acceptance probability of these expensive updates can be low. This paper provides a generalization of Møller et al. (2004) and a new MCMC algorithm, which obtains better acceptance probabilities for the same amount of exact sampling, and removes the need to estimate model parameters before sampling begins.

*Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence (UAI)*, 2006.
[PDF, DjVu, GoogleViewer, BibTeX]

**See also:** Chapter 5 of my PhD thesis
includes more detail, context, and interpretations.