Bayesian Nonparametric Hypothesis Testing for Complex Data

Abstract

In this thesis, we propose novel Bayesian Nonparametric (BNP) hypothesis testing procedures for four familiar problems where parametric assumption are untenable. Out first proposed is a BNP hypothesis testing procedure for comparing the multivariate distributions of several treatment groups against a control group. This test is derived from a flexible model for the group distributions based on a random binary vector such that, if its jth element equals one, then the jth treatment group is merged with the control group. The group distributions’ flexibility comes from a dependent Dirichlet process, while the latent vector prior distribution ensures a multiplicity correction to the testing procedure. We explore the posterior consistency of the Bayes factor and provide a Monte Carlo simulation study comparing the performance of our procedure with state-of-the-art alternatives. The second proposal is a BNP hypothesis testing procedure for comparing several treatments against a control, but now in a general univariate ANOVA designs. Specifically, our strategy considers the definition of latent binary variables, which identify each cell with different random distributions. Such distributions are modeled via infinite mixture models, where the mixing distributions follow a dependent Dirichlet process with shared weights and a base distribution that depends on the type of data: Gaussian for continuous data, Bernoulli-Poisson for count data, and Bernoulli for binary data. Then, the prior distribution on the hypothesis space is defined with a distribution over the binary vector. In summary, we relax the classical ANOVA assumptions and propose models for testing the general ANOVA hypotheses considering data in different supports. The third proposal is a BNP hypothesis testing procedure for evaluating the Granger causality between two or more time series. Our test is based on a flexible model for the joint evolution of multiple series, where a latent binary matrix indicates whether there is a Granger-causal relationship between such time series. Our model is specified through a dependent Geometric stick-breaking process that generalizes the standard parametric Gaussian vector autoregression model, whereas the prior distribution of the latent matrix ensures a multiple testing correction. We provide a Monte Carlo simulation study comparing the robustness of our hypothesis test with state-of-the-art alternatives. Finally, our fourth proposal is a BNP model for density regression and model selection. The conditional distributions are flexibly modeled using a dependent stick-breaking process, and the model selection is achieved using a Dirac spike-and-slab prior. Our model maintains the attractive properties of the geometric stick-breaking process and leverages in recent Polya-Gamma and Chinese restaurant table data augmentations, created for parametric logistic and negative binomial regression, to develop an efficient Gibbs sampler. We test the performance of our model selection procedure through simulation studies. Most moderate sample sizes, our algorithm works better than other popular parametric model selection procedures.