Nonparametric Bayesian Models for Markov Jump Processes

Markov jump processes (MJPs) have been used as models in various fields such as disease progression, phylogenetic trees, and communication networks. The main motivation behind this thesis is the application of MJPs to data modeled as having complex latent structure. In this thesis we propose a nonparametric prior, the gamma-exponential process (GEP), over MJPs. Nonparametric Bayesian models have recently attracted much attention in the statistics community, due to their flexibility, adaptability, and usefulness in analyzing complex real world datasets. The GEP is a prior over infinite rate matrices which characterize an MJP; this prior can be used in Bayesian models where an MJP is imposed on the data but the number of states of the MJP is unknown in advance. We show that the GEP model we propose has some attractive properties such as conjugacy and simple closed-form predictive distributions. We also introduce the hierarchical version of the GEP model; sharing statistical strength can be considered as the main motivation behind the hierarchical model. We show that our hierarchical model admits efficient inference algorithms. We introduce two inference algorithms: 1) a “basic” particle Markov chain Monte Carlo (PMCMC) algorithm which is an MCMC algorithm with sequences proposed by a sequential Monte Carlo (SMC) algorithm; 2) a modified version of this PMCPC algorithm with an “improved” SMC proposal. Finally, we demonstrate the algorithms on the problems of estimating disease progression in multiple sclerosis and RNA evolutionary modeling. In both domains, we found that our model outperformed the standard rate matrix estimation approach.