Bayesian model determination for multivariate ordinal and binary data

Different conditional independence specifications for ordinal categorical data are compared by calculating a posterior distribution over classes of graphical models. The approach is based on the multivariate ordinal probit model where the data are considered to have arisen as truncated multivariate normal random vectors. By parameterising the precision matrix of the associated multivariate normal in Cholesky form, ordinal data models corresponding to directed acyclic conditional independence graphs for the latent variables can be specified and conveniently computed. Where one or more of the variables are binary this parameterisation is particularly compelling, as necessary constraints on the latent variable distribution can be imposed in such a way that a standard, fully normalised, prior can still be adopted. For comparing different directed graphical models a reversible jump Markov chain Monte Carlo (MCMC) approach is proposed. Where interest is focussed on undirected graphical models, this approach is augmented to allow switches in the orderings of variables of associated directed graphs, hence allowing the posterior distribution over decomposable undirected graphical models to be computed. The approach is illustrated with several examples, involving both binary and ordinal variables, and directed and undirected graphical model classes.