| Summary: | Indiana University-Purdue University Indianapolis (IUPUI) === The essence of Bayesian data analysis is to ascertain posterior distributions. Posteriors
generally do not have closed-form expressions for direct computation in practical applications.
Analysts, therefore, resort to Markov Chain Monte Carlo (MCMC) methods for the generation
of sample observations that approximate the desired posterior distribution. Standard MCMC
methods simulate sample values from the desired posterior distribution via random proposals.
As a result, the mechanism used to generate the proposals inevitably determines the
efficiency of the algorithm. One of the modern MCMC techniques designed to explore
the high-dimensional space more efficiently is Hamiltonian Monte Carlo (HMC), based on
the Hamiltonian differential equations. Inspired by classical mechanics, these equations
incorporate a latent variable to generate MCMC proposals that are likely to be accepted.
This dissertation discusses how such a powerful computational approach can be used for
implementing statistical models. Along this line, I created a unified computational procedure
for using HMC to fit various types of statistical models. The procedure that I proposed can
be applied to a broad class of models, including linear models, generalized linear models,
mixed-effects models, and various types of semiparametric regression models. To facilitate
the fitting of a diverse set of models, I incorporated new parameterization and decomposition
schemes to ensure the numerical performance of Bayesian model fitting without sacrificing
the procedure’s general applicability. As a concrete application, I demonstrate how to use the
proposed procedure to fit a multivariate generalized additive model (GAM), a nonstandard
statistical model with a complex covariance structure and numerous parameters. Byproducts of the research include two software packages that all practical data analysts to use the
proposed computational method to fit their own models. The research’s main methodological
contribution is the unified computational approach that it presents for Bayesian model
fitting that can be used for standard and nonstandard statistical models. Availability of
such a procedure has greatly enhanced statistical modelers’ toolbox for implementing new
and nonstandard statistical models.
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