Nonparametric Bayesian statistics harnessing the forces of data in change-point detection and survival analysis

Abstract

Bayesian nonparametric priors are distributions on functions. In this thesis, we present several novel Bayesian approaches based on the elicitation of a set of nonparametric priors in two problems, change-point detection, and survival analysis. Through our success on each target, we demonstrate the fact that appropriate Bayesian nonparametric priors can harness the power of the data and promote statistical analysis from the perspectives of estimation, inference, prediction, and computation. In Part I, we propose NOSE and SBPCPM, two jump-size-based Bayesian approaches to solve change-point detection. NOSE globally models the abrupt change process and identifies change-points based on the induced posterior estimates of jump sizes. We establish posterior inferential theories including the minimax optimality of posterior contraction, posterior consistency of both number and locations of change-points, and an asymptotic zero false negative rate in change-point discrimination under a novel Gamma-IBP weighted spike-and-slab type prior. Comprehensive numerical studies demonstrate that NOSE outperforms existing approaches. SBPCPM is extremely useful to detect the imperceptible change-points under a mean-shifted model. We propose a novel Beta process mixture model for the change signal. We establish the pointwisely asymptotic efficiency of the marginal MAP estimates of the change signal under the hypothesis of no change-points. The induced asymptotic normality of the jump size estimators leads to efficient hypothesis testing of change-points. In Part II, we study the use of nonparametric priors in survival analysis. For right-censored survival outcomes, we propose BuLTM, a novel Bayesian method for prediction under the non-parametric transformation model. Unlike existing methods, we allow the model to be unidentified and assign weakly informative nonparametric priors to the infinite-dimensional parameters to facilitate efficient MCMC sampling. We show that the posterior is proper under the unidentified model. For recurrent event data, we propose a generalized shared frailty model to relax the strict proportional hazard assumption and apply the ANOVA DPP as the prior for baseline survival functions for model estimation.