http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Multivariate Normal Probability, Truncated Sampling, and Approximation for Statistical Modeling
This dissertation contains four projects involving applications of truncated multivariate normal sampling and multivariate normal probability estimation for linearly constrained domains. These two problems have a large number of applications in various statistical modeling domains. They are also very challenging to solve efficiently and accurately for small probability domains, which inevitably occurs as the dimension of the multivariate normal increases. Thus models that depend on such components demand scalable and accurate samplers and probability estimation methods. The first project is a survey of truncated multivariate normal sampling and multivariate normal probability estimation. We provide exposition on commonly used samplers and probability estimation methods in the literature, and trace their development. We provide C++ implementations of important methods, and compare them on synthetic and realistic examples to gauge their scalability and accuracy. The second project is an application of expectation propagation for multivariate normal approximation to multinomial probit modeling with sparsity constraints on the latent precision. We use the approximation to compute moments of truncated multivariate normals for an expectation maximization algorithm. The third project is an application of expectation propagation to a weighted Gaussian graphical model for heterogenous graph modeling. We explore using the full covariance multivariate normal approximation and inclusive Kullback Leibler divergence objective of expectation propagation and find improved performance compared to a simpler mean field normal variational inference approximation. The final project introduces a measure for quantifying the relative effective sample size between Gaussian processes with different covariance kernels. This provides insight on an important hyperparameter for Gaussian process regression. The electronic version of this dissertation is accessible from https://hdl.handle.net/1969.1/198024
Doctor of Philosophy Texas A&M University 2022 https://hdl.handle.net/1969.1/198024