| Popis: |
In Bayesian nonparametrics, the specification of suitable (for practical purposes) stochastic processes whose realisations are discrete probability measures plays a crucial role. Recently, real world applications have motivated the extension of these stochastic processes to incorporate covariate information in the realisations with the aim of constructing infinite mixture models having weights and/or component-specific parameters which depend on covariates. This work presents four different modelling strategies motivated by practical problems involving stochastic processes over covariate dependent random measures. After presenting the main concepts in Bayesian nonparametrics and reviewing relevant literature, we develop two Bayesian models which are extensions of augmented response mixture models. In particular, we construct a semi-parametric non-linear regression model for zero-inflated discrete distributions and propose techniques to perform variable selection in cluster-specific regression models. The third contribution presents a generalisation of Dirichlet Process for random probability measures to include covariate information via Beta regression. Properties of this new stochastic process are discussed and two illustrations are presented for dealing with spatially correlated observations and grouped longitudinal data. The last part of the thesis proposes a modelling strategy for time-evolving correlated binary vectors, which relies on latent variables. The distribution of these latent variables is assumed to be a convolution of Gaussian kernels with covariate dependent random probability measures. These four modelling strategies are motivated by datasets that come from medical studies involving lower urinary tract symptoms and acute lymphoblastic leukaemia as well as from publicly available data about primary schools evaluations in London. |
