Mixture models are widely used in many application areas, with finite mixtures of Gaussian distributions applied routinely in clustering and density estimation. With the increasing need for a flexible model for predictor-dependent clustering and conditional density estimation, mixture models are generalized to incorporate predictors with infinitely many components in the semiparametric Bayesian perspective. Much of the recent work in the nonparametric Bayes literature focuses on introducing predictor-dependence into the probability weights.

In this dissertation we propose three semiparametric Bayesian methods, with a focus on the applications of predictor-dependent clustering and condition density estimation. We first derive a generalized product partition model (GPPM), starting with a Dirichlet process (DP) mixture model. The GPPM results in a generalized Pólya urn scheme. Next, we consider the problem of density estimation in cases where predictors are not directly measured. We propose a model that relies on Bayesian approaches to modeling of the unknown distribution of latent predictors and of the conditional distribution of responses given latent predictors. Finally, we develop a semiparametric Bayesian model for density regression in cases with many predictors. To reduce dimensionality of data, our model is based on factor analysis models with the number of latent variables unknown. A nonparametric prior for infinite factors is defined.