Bayesian methodologies for constrained spaces.

Due to advances in technology, there is a presence of directional data in a wide variety of fields. Often distributions to model directional data are defined on manifold or constrained spaces. Regular statistical methods applied to data defined on special geometries can give misleading results, and this demands new statistical theory. This dissertation addresses two such problems and develops Bayesian methodologies to improve inference in these arenas. It consists of two projects: 1. A Bayesian Methodology for Estimation for Sparse Canonical Correlation, and 2. Bayesian Analysis of Finite Mixture Model for Spherical Data. In principle, it can be challenging to integrate data measured on the same individuals occurring from different experiments and model it together to gain a larger understanding of the problem. Canonical Correlation Analysis (CCA) provides a useful tool for establishing relationships between such data sets. When dealing with high dimensional data sets, Structured Sparse CCA (ScSCCA) is a rapidly developing methodological area which seeks to represent the interrelations using sparse direction vectors for CCA. There is less development in Bayesian methodology in this area. We propose a novel Bayesian ScSCCA method with the use of a Bayesian infinite factor model. Using a multiplicative half Cauchy prior process, we bring in sparsity at the level of the projection matrix. Additionally, we promote further sparsity in the covariance matrix by using graphical horseshoe prior or diagonal structure. We compare the results for our proposed model with competing frequentist and Bayesian methods and apply the developed method to omics data arising from a breast cancer study. In the second project, we perform Bayesian Analysis for the von Mises Fisher (vMF) distribution on the sphere which is a common and important distribution used for directional data. In the first part of this project, we propose a new conjugate prior for the mean vector and concentration parameter of the vMF distribution. Further we prove its properties like finiteness, unimodality, and provide interpretations of its hyperparameters. In the second part, we utilize a popular prior structure for a mixture of vMF distributions. In this case, the posterior of the concentration parameter consists of an intractable Bessel function of the first kind. We propose a novel Data Augmentation Strategy (DAS) using a Negative Binomial Distribution that removes this intractable Bessel function. Furthermore, we apply the developed methodology to Diffusion Tensor Imaging (DTI) data for clustering to explore voxel connectivity in human brain