Bayesian model choice for multivariate ordinal data

This thesis provides a coherent and adaptable methodology for multivariate ordinal and binary data. Two main aspects of data modelling are considered. The first is to formulate a model for the data and to estimate the model parameters using Bayesian computation. The second is to assess model choice; models considered are the set of directed acyclic graphical models and the set of decomposable models. The model is based on the multivariate probit model (Chib and Greenberg, 1998) but parameterised in a way that makes computation convenient. In particular, the conditional posterior distributions of the model parameters are standard and easily simulated from using Gibbs sampling techniques. Prior parameters are chosen to be noninformative but not overly diffuse. The Gibbs sampler is applied successfully to examples, and the goodness-of-fit of the model is assessed using simulation techniques. The model parameterisation allows ordinal and binary data and a mixture of both data types to be modelled within the same framework. Reversible Jump Markov chain Monte Carlo methods are used to estimate posterior model probabilities for directed acyclic graphical models. Under the model parameterisation described, a suitable proposal distribution is easily specified. The issue of model choice is also investigated for the set of (undirected) decomposable models. Under some model parameterisations, the conditional independence structure of a decomposable model can not be specified. A further Reversible Jump Markov chain Monte Carlo step is described to move between model parameterisations. Both Reversible Jump algorithms are found to rapidly explore the model and parameter spaces. The model is extended for data where covariates are also observed. The Reversible Jump algorithms described previously are adapted and applied to examples. A further Reversible Jump step is developed and implemented to assess which covariates should be included in a model to predict the data.