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Bayesian Model Determination for Multivariate Ordinal and Binary Data

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We consider how to compare different conditional independence specifications for ordinal categorical variables, by calculating a posterior distribution over classes of graphical models. The approach is based on the multivariate ordinal probit model (Chib and Greenberg, 1998) 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 (e.g. as Smith and Kohn, 2002) ordinal data models corresponding to directed acyclic conditional independence graphs can be specified and conveniently computed. Where one or more of the variables is 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 we propose a reversible jump MCMC approach. 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.

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