Gilks, W. R. (ed.) et al., Markov chain Monte Carlo in practice. London: Chapman & Hall. 215-239 (1996).
In recent years, the use of Markov Chain Monte Carlo (MCMC) simulation techniques has made feasible the routine Bayesian analysis of many complex high-dimensional problems. However, one area which has received relatively little attention is that of comparing models of possibly different dimensions, where the essential difficulty is that of computing the high-dimensional integrals needed for calculating the normalization constants for the posterior distribution under each model specification.
Here, we show how methodology developed recently by U. Grenander and M. I. Miller [see J. R. Stat. Soc., Ser. B 56, No. 4, 549-603 (1994; Zbl 0814.62009); Tech. Rep., Electr. Sig. Syst. Res. Lab., Washington Univ. (1991)] can be exploited to enable routine simulation-based analyses for this type of problem. Model uncertainty is accounted for by introducing a joint prior probability distribution over both the set of possible models and the parameters of those models. Inference can then be made by simulating realizations from the resulting posterior distribution using an iterative jump-diffusion sampling algorithm. The essential features of this simulation approach are that discrete transitions, or jumps, can be made between models of different dimensionality, and within a model of fixed dimensionality the conditional posterior appropriate for that model is sampled. Model comparison or choice can then be based on the simulated approximation to the marginal posterior distribution over the set of models.