On some computational, modeling and design issues in Bayesian analysis of spatial data

Abstract

My research on Bayesian spatial analysis can be divided into three challenges: computing, methodology (modeling) and experimental design. My first exploration in research is to find an alternative to Markov chain Monte Carlo (MCMC) for the Bayesian hierarchical model. Variational Bayesian (VB) method would be a choice to tackle the massive computational burden for large spatial data analysis. We discuss applying VB to spatial analysis, especially to the multivariate spatial cases. Different VB algorithms are developed and applied to simulated and real examples. When the number of the locations and the dimension of the outcome variables are large, models with feature of dimension reduction are essential in the real applications. Low-rank spatial processes and factor analysis models are merged together to capture the associations among the variables as well as the strength of spatial correlation for each variable. We also develop stochastic selection of the latent factors by utilizing certain identifiability characterizations for the spatial factor model. A MCMC algorithm is developed for estimation, which also deals with the spatial misalignment problem. In many of the spatial applications (environmental epidemiology, for instance), parameter estimation is the most important objective in the study. Even with carefully constructed models and computing technique, it is always a challenge to handle the large spatial data set. Bayesian experimental design may help us to get the desired information from a spatial survey study with a sample size that can be analyzed by most available software. The problem of finding the optimum experimental design for the purpose of performing one or more hypothesis tests is considered in the context of spatial analysis. The Bayesian decision theoretic approach is used to arrive at several new optimality criteria for this purpose. Different approaches to achieving this goal are explored, including additive weighted loss and convex approximation. Simulated annealing algorithm (SAA) is applied to real examples to find the optimum design based on our objective function.