  The reversible jump Markov chain Monte Carlo (RJMCMC) method offers an
across-model simulation approach for Bayesian estimation and model comparison,
by exploring the sampling space that consists of several models of possibly
varying dimensions. A naive implementation of RJMCMC to models like Gibbs
random fields suffers from computational difficulties: the posterior
distribution for each model is termed doubly-intractable since computation of
the likelihood function is rarely available. Consequently, it is simply
impossible to simulate a transition of the Markov chain in the presence of
likelihood intractability. A variant of RJMCMC is presented, called noisy
RJMCMC, where the underlying transition kernel is replaced with an
approximation based on unbiased estimators. Based on previous theoretical
developments, convergence guarantees for the noisy RJMCMC algorithm are
provided. The experiments show that the noisy RJMCMC algorithm can be much more
efficient than other exact methods, provided that an estimator with controlled
Monte Carlo variance is used, a fact which is in agreement with the theoretical
analysis.
