Efficient Model Determination for Discrete

A graphical model is a family of probability distributions incorporating the conditional independence assumptions represented by a graph. They are constructed by specifying local dependencies of each node of the graph in terms of its immediate neighbours. It is then possible to work locally, obtaining better results in terms of statistical inference and computational eeciency. Our motivation here is to develop eecient procedures for Bayesian model determination in discrete graphical models, employed for the analysis of contingency tables. For high-dimensional contingency tables the set of plausible models is large, and a full comparison of all the posterior probabilities associated to the competing models becomes infeasible. In fact the number of graphical structures to examine increases more than exponentially with the number of vertices. Various solutions to this problem have been proposed, the one we suggest is based on the application of MCMC techniques. This possibility has already been exploited. Madigan and York (1995), for example, introduce an MCMC sampler, called Markov chain Monte Carlo composition (MC 3 hereafter), for the analysis of decomposable models. They construct a Metropolis Hastings sampler that permits to explore the space of all decomposable models. Alternatively, Dellaportas and Foster (1999) have developed a MCMC sampler for model choice in loglinear models which, although including discrete graphical models, does not require decomposability and, therefore, does not lead to local computations, necessary to analyse large domains. We shall present two diierent MCMC samplers for the analysis of decomposable discrete graphical models, which are fully based on local computations and, therefore, eecient. The rst one is a revised version of the MC 3 algorithm by Madigan and York (1995). It diiers from the original version mainly because it incorporates a local condition for checking decomposability. Furthermore, we shall propose an extension which allows for a hierarchical prior on the cell counts. The second sampler is based on the Reversible jump JMCMC by Green (1995). The methodology proposed parallels that presented in Giudici and Green (1999) for the analysis of decomposable gaussian models. As in the gaussian case, at each step of the algorithm we update not only the graphical structure (as in MC 3), but also the associated parameter vector. Otherwise, there are substantial diierence due to the data structure. Essentially, in the gaussian case, pairwise conditional independence is dictated by the absence of a single parameter, whereas in the discrete case this corresponds in general to non …