UC Riverside

Markov chain Monte Carlo (MCMC) methods are highly desirable when the sampling distribution is intractable. Among all MCMC methods, the fundamental one is the Metropolis-Hastings algorithm. Despite its extensive application in approximating any distribution, the Markov chain often suffers from slow mixing, which then causes insufficient estimation. We address this issue by proposing modifications to the Metropolis-Hastings algorithm that, under specified conditions, induces substantial improvements in jump distances and statistical efficiency while preserving the overall quality of convergence. This dissertation starts with an introduction of the MCMC methods and continues by proposing the Efficient Conditional Metropolis-Hastings algorithm (ECMH) and a variation of ECMH under a uniform setting (ECMHu). We further investigate their properties through a series of models, including a Bivariate normal model, a Bayesian random effects model, and a Bayesian dynamic spatiotemporal model. Simulation results are compared across all algorithms.

Spatiotemporal processes are ubiquitous in the environmental and physical sciences. The complexity of these processes and a large number of observations preclude the use of traditional models such as partial differential equations, integrodifference equations, and covariance based space-time models. Alternatively, the spatiotemporal hierarchical Bayesian models are ideal in this case as it can conditionally specify the components in the model and eventually link them together through Bayes' Theorem. However, the complex and high-dimensional nature of these models prevents the direct evaluation of the posterior distribution. Instead, we can apply MCMC methods to draw samples from the posterior distribution and make Bayesian inferences. In fact, MCMC methods have revolutionized such modeling by allowing for more realistic and complicated models. As a novel application of the MCMC methods, we propose several spatiotemporal Hierarchical Bayesian models to understand the dynamic of post-fire chaparral recovery with data collected from the Angeles National Forest. This dissertation continues to investigate a particular spatiotemporal process of galaxy formation and evolution, in which the environment (cosmic web) plays a major role. However, the relation between galaxies and environment is not well understood. To this end, we propose a multi-step approach of representing galaxy formation trees as feature vectors and classifying along with galaxy properties to the environment.