Exploration of Densities from a Bayesian Perspective

Abstract

This dissertation focuses on solving some of the most interesting theoretical and methodological questions arising out of various different disciplines with a Bayesian perspective. With the advent of large scale dataset and the ability to crunch through it, there is a dire need for scalable Bayesian methodologies. Variational Bayes (VB) techniques has become hugely popular feeding into that void. The first chapter analyses a popular VB technique namely, Tangent Transform (TT) approach. We take a comprehensive look at the technique and provide theoretical guarantees for the algorithm convergence, as well as inspecting it’s associated risk bounds. VB often approximates an unknown density with a density from a known class (e.g. Gaussian) via a ‘close match’ (Kullback–Leibler divergence) metric. The second chapter is exploration of associated risk bounds in Gaussian approximation of density belonging to generalized linear models (GLM).

The third chapter is devoted to estimation of conditional density in a high-dimensional setup with spatially varying responses. We develop robust model to produce viable inference on the data as well as theoretical results on posterior contraction.