Date of Award

2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Statistics

First Advisor

Edward I. George

Abstract

This dissertation explores Bayesian model selection and estimation in settings where the model space is too vast to rely on Markov Chain Monte Carlo for posterior calculation. First, we consider the problem of sparse multivariate linear regression, in which several correlated outcomes are simultaneously regressed onto a large set of covariates, where the goal is to estimate a sparse matrix of covariate effects and the sparse inverse covariance matrix of the residuals. We propose an Expectation-Conditional Maximization algorithm to target a single posterior mode. In simulation studies, we find that our algorithm outperforms other regularization competitors thanks to its adaptive Bayesian penalty mixing. In order to better quantify the posterior model uncertainty, we then describe a particle optimization procedure that targets several high-posterior probability models simultaneously. This procedure can be thought of as running several ``mutually aware'' mode-hunting trajectories that repel one another whenever they approach the same model. We demonstrate the utility of this method for fitting Gaussian mixture models and for identifying several promising partitions of spatially-referenced data. Using these identified partitions, we construct an approximation for posterior functionals that average out the uncertainty about the underlying partition. We find that our approximation has favorable estimation risk properties, which we study in greater detail in the context of partially exchangeable normal means. We conclude with several proposed refinements of our particle optimization strategy that encourage a wider exploration of the model space while still targeting high-posterior probability models.

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