A Beta Cauchy-Cauchy (BECCA) prior for sparse signal recovery in regression and graphical models.

Abstract

This PhD thesis introduces a novel Bayesian approach for variable selection in high- dimensional regression settings, along with its potential extension to learning the structure of an undirected graphical model. Our proposed method, which we call the Beta Cauchy-Cauchy (BECCA) prior, replaces the indicator variables in the traditional spike and slab prior with continuous, Beta-distributed random variable and places half-Cauchy priors over the para-meters of the Beta distribution, which significantly improves the predictive and inferential performance of the technique. Similar to shrinkage methods, our continuous analog of the Spike-and-Slab (SS) prior enables posterior exploration using gradient-based methods, such as Hamiltonian Monte Carlo (HMC), while at the same time explicitly allowing for variable selection in a principled Bayesian framework. Building on the strong performance of the proposed approach in linear regression, we apply it to logistic regression context and further extend it to structure learning in Gaussian Graphical Models (GGMs) using a regression based framework. We evaluate the frequentist properties of our model through simulations and demonstrate that our technique not only outperforms the latest Bayesian variable selection methods in linear regression, but also performs comparably or better than existing methods for variable selection in logistic regression and structure learning in graphical models. The efficacy, applicability and performance of our approach, are further underscored through its implementation on real datasets.