Bayesian Hierarchical Models for Multivariate Continuous and Categorical Data

Abstract

This thesis explores the use of the scale mixtures of normal (SMN) family of probability distributions as a data augmentation strategy in various Bayesian models for continuous and categorical data. The purpose is to facilitate efficient Bayesian computational methods, where Markov chain Monte Carlo (MCMC) algorithms have been a standard choice for handling sophisticated models. First, this thesis considers the use of the modified multivariate Student-t (Mod-t) distribution in seemingly unrelated regression (SUR) models. The Mod-t distribution allows for flexibly modelling the tails of its independent marginal Student-t distributions. Although the probability density function (PDF) of the Mod-t distribution does not have a closed form, it can be expressed into an SMN representation. This simplifies the Gibbs sampler for the SUR modelling. The applications to the Kenneth French data and the Dominick’s Finer Foods retail sales data have shown promising results, and these are compared to the Gaussian and Student-t copulas. Second, this thesis proposes a new class of scale mixtures of skew-normal distributions, which is the modified multivariate skew-t (Mod-skew-t) distribution. The statistical properties of this new probability distribution are explored. An application to the multivariate GARCH model with Mod-skew-t innovations to US stock returns is illustrated. Third, this thesis also studies the views and attitudes of Australian voters in the 2016 Australian Election Study survey using latent class regressions. The probabilities for the latent classes are assumed to follow a multinomial logit link, which has an SMN representation with Pólya-Gamma latent variables. Covariates for the class probabilities are selected via a spike and slab prior for the regression coefficients. The hierarchical representation of this model leads to an efficient Gibbs sampler.