Bayesian nonparametric regression via partitioning

Bayesian inference offers us a powerful tool with which to tackle the problems of data modelling. However the performance of Bayesian methods is crucially dependent on being able to find good models for our data. The principal focus of this thesis is the development of models based on the Bayesian Partition Model. Such models have the flexibility to model complex phenomena while being mathematically simple.;Smoothing methods have been an active area of research since the late 1980s. In this thesis, I present a new Bayesian approach for smoothing to predict on a continuous covariate space. The Bayesian Partition Models construct a complex arbitrary regression or classification surface over the space by splitting the domain into a data-determined number of regions. Within each region the data is assumed to be homogeneous and come from simple distributions. Under conjugate priors within each region we can obtain the marginal likelihood analytically. Markov Chain Monte Carlo (MCMC) methods are used to sample the posterior distributions of models. The predictive distributions can be obtained by averaging across the models.;Two MCMC methods for transdimensional sampling problems are explored. One is reversible-jump MCMC (RJMCMC) and the other is continuous time birth-death MCMC (BDMCMC).;I conclude with directions for further research and a package for the R system.