Bayesian Computations via MCMC, with applications to Big Data and Spatial Data

Markov Chain Monte Carlo (MCMC) methods are fundamental tools for sampling highly complex distributions. They are crucial to Bayesian inference as posterior distributions are generally analytically intractable. In this thesis, we tackle two Bayesian inference problems via MCMC methods, that will lie on both methodology and application aspects.\nThe first part of this thesis tackles the computational challenges of Bayesian inference from big data. We develop a new communication-free parallel method, the "Likelihood Inflating Sampling Algorithm (LISA)", that significantly reduces computational costs by randomly splitting the dataset into smaller subsets and running MCMC methods independently in parallel on each subset using different processors. Each processor will be used to run an MCMC chain that samples sub-posterior distributions which are defined using an "inflated" likelihood function. We then discuss on the approaches to combine all sub-samples from all processors to build a highly accurate posterior distribution that is consistent with the full posterior distribution. More importantly, we learn a strategy in combining LISA's draws to study the full posterior of the more complex Bayesian Additive Regression Trees (BART) model, which is highly important in non-parametric regression. We also successfully examine the consistency in performance of LISA on BART with new efficient Metropolis-Hastings proposals introduced by Pratola (2016).\nThe second part of this thesis is focused on the applied aspect of performing Bayesian inference with MCMC methods. We study a Bayesian Geostatistical model to analyze spatial data from the Timiskaming Abitibi River forests in Ontario Canada, provided by the First Resource Management Group Inc.. We implement an MCMC algorithm to perform Bayesian inference on predicting the proportion of hardwood trees from elevation and vegetation index. Spatial predictions are made for new sites in the forests and results are compared with a Logistic Regression model without a spatial effect. We study the trend of accuracy in predictions when fitting fewer data to the model, and present useful insights on the trade-off between performance and the costly need for collecting ground truth data. We further discuss a stratified sampling approach in choosing the subsets of data that allows for potential better predictions.

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