Bayesian Density Regression and Predictor-Dependent Clustering

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 Polya 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

Generalized Negative Binomial Processes and the Representation of Cluster Structures

The paper introduces the concept of a cluster structure to define a joint distribution of the sample size and its exchangeable random partitions. The cluster structure allows the probability distribution of the random partitions of a subset of the sample to be dependent on the sample size, a feature not presented in a partition structure. A generalized negative binomial process count-mixture model is proposed to generate a cluster structure, where in the prior the number of clusters is finite and Poisson distributed and the cluster sizes follow a truncated negative binomial distribution. The number and sizes of clusters can be controlled to exhibit distinct asymptotic behaviors. Unique model properties are illustrated with example clustering results using a generalized Polya urn sampling scheme. The paper provides new methods to generate exchangeable random partitions and to control both the cluster-number and cluster-size distributions.Comment: 30 pages, 8 figure