Flexible Bayesian modelling of concomitant covariate effects in mixture models

Abstract

Mixture models provide a useful tool to account for unobserved heterogeneity, and are the basis of many model-based clustering methods. In order to gain additional flexibility, some model parameters can be expressed as functions of concomitant covariates. In particular, prior probabilities of latent group membership can be linked to concomitant covariates through a multinomial logistic regression model, where each of these so-called component weights is associated with a linear predictor involving one or more of these variables. In this Thesis, this approach is extended by replacing the linear predictors with additive ones, where the contributions of some/all concomitant covariates can be represented by smooth functions. An estimation procedure within the Bayesian paradigm is proposed. In particular, a data augmentation scheme based on difference random utility models is exploited, and smoothness of the covariate effects is controlled by suitable choices for the prior distributions of the spline coefficients. This methodology is then extended to include flexible covariates effects also on the component densities. The performance of the proposed methodologies is investigated via simulation experiments and applications to real data. The content of the Thesis is organized as follows. In Chapter 1, a literature review about mixture models and mixture models with covariate effects is provided. After a brief introduction on Bayesian additive models with P-splines, the general specification for the proposed method is presented in Chapter 2, together with the associated Bayesian inference procedure. This approach is adapted to the specific case of categorical and continuous manifest variables in Chapter 3 and Chapter 4, respectively. In Chapter 5, the proposed methodology is extended to include flexible covariate effects also in the component densities. Finally, conclusions and remarks on the Thesis are collected in Chapter 6.