Bayesian stochastic search for Ising chain graph models

A class of Ising chain graph models is illustrated to explore the effect of an external risk factor on a set of binary outcomes and on their joint dependence structure modelled via undirected graphs, where we are mainly interested in the risk factor effect on pairwise associations between outcomes rather on single outcomes. Under the LWF Markov property, the joint probability mass associated to a chain graph corresponds to a log-linear model with suitable zero constraints in correspondence of missing edges. We devise a Bayesian Ising model based on conjugate priors for log-linear parameters that aims at the selection of the best graph that fits the data. A computational strategy is implemented that uses Laplace approximations and a Metropolis-Hastings algorithm that allows us to perform model selection.