Bayesian Estimation and Model Selection for Single and Multiple Graphical Models

• Main Author: Jalali, Peyman
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2019
• Subjects: Applied Mathematics; Statistical physics; Statistics
• ISBN: 9798691293597

Undirected Graphical Models represent a family of canonical statistical models for reconstructing interactions amongst a set of entities from multi-dimensional data profiles. They have numerous applications in biology involving Omics and neuroimaging data, in social sciences for voting records and econ/financial data, in text mining, etc. Recently, the problem of joint estimation of multiple graphical models from high dimensional data has also received much attention in the statistics and machine learning literature, due to its importance in diverse fields including molecular biology, neuroscience and the social sciences. In the first part of this dissertation, we will develop two Bayesian methodologies, using spike and slab and continuous shrinkage priors coupled with a pseudo-likelihood that enables fast computations, for estimating a single high-dimensional graphical model. We will introduce efficient Gibbs samplers and illustrate the efficiency of the models by comparing with the state of the art models. The second part develops a Bayesian approach that decomposes the model parameters across multiple underlying graphical models into shared components across subsets of models and edges, and idiosyncratic ones. Further, it leverages a novel multivariate prior distribution, coupled with the same pseudo-likelihood as above through a robust and efficient Gibbs sampling scheme. We establish strong posterior consistency for model selection, as well as estimation of model parameters under high dimensional scaling with the number of variables growing exponentially with the sample size. The efficacy of the proposed approach is illustrated on both synthetic and real data.