Bayesian inference for binary regression model and hierarchical copula models

Statistical inference provides a framework for understanding the true data-generating process by analyzing a subset of observed data. Central to any inferential approach is the choice of the statistical model, which influences both the interpretation and the methodology of inference. This thesis focuses on the Bayesian paradigm, a cornerstone of modern statistical inference, and explores its application to binary regression models and hierarchical copula models. The first part of the thesis is dedicated to Bayesian binary regression. We begin by presenting an acceptance-rejection algorithm for random number generation from the Kolmogorov distribution. The derivation relies on proving uniform convergence of the derivative series expansion and establishing an optimal truncation method for the density function. We then introduce a new class of distributions, the Perturbed Unified Skew Normal (PUSN), which extends the SUN family and serves as a conjugate prior for Bayesian binary regression models with link functions expressible as scale mixtures of Gaussian densities. A detailed study of the logit case demonstrates that posterior summaries, such as quantiles and normalizing constants, can be efficiently computed. We propose a Gibbs sampler for cases involving more general priors and show its superior performance in high-dimensional settings (p > n). The efficacy of the proposed methods is demonstrated through simulations and real-world datasets. Furthermore, we introduce the Extended Quasi SUN distribution, a novel generalization of the SUN family, and establish its utility in Bayesian nonparametric binary regression. We define the Extended Quasi SUN process, prove its conjugacy properties for a wide range of link functions, and develop an importance sampling algorithm for posterior computation. The second part of the thesis explores Bayesian hierarchical copula models. We employ the Dirichlet-Laplace prior to address limitations in existing approaches that may yield improper posteriors. By incorporating a global–local shrinkage prior, we account for potential dependence structures among different groups. The performance of the hierarchical copula model is evaluated through extensive simulations and real data applications. Beyond model development, this thesis addresses key objectives of statistical inference, including parameter estimation, model selection, and predictive analysis. Chapter 1 focuses on computational aspects of the Kolmogorov distribution, which play a fundamental role in the algorithms developed in subsequent chapters. Chapter 2 delves into parameter estimation and model selection for parametric logit and probit models. Chapter 3 extends these concepts to prediction in nonparametric logistic regression. Finally, Chapter 4 shifts focus to hierarchical copula models, emphasizing parameter estimation and dependence structure analysis. Through these contributions, this thesis advances Bayesian statistical methodology and enhances computational techniques for complex inferential problems.