Bayesian Computational Statistics

A rigorous introduction to the theory of Bayesian Statistical Inference and Data Analysis, including prior and posterior distributions, Bayesian estimation and testing, Bayesian computation theories and methods, and implementation of Bayesian computation methods using popular statistical software. Required Textbook: Gelman, A., Carlin, J. B., Stern, H. S., Rubin, D. B. (2013) Bayesian Data Analysis, Third Edition, Chapman & Hall/CRC. Software Requirements: R or Python, Word processing (such as Word, Pages, LaTeX, etc)

• Module 1: Fundamentals of Bayesian Inference
• Welcome to MATH 574 Bayesian Computational Statistics! This module covers the ideas of Bayesian inference. It focuses on a framework for Bayesian inference and discusses the general approach to computation.
• Module 2: Single Parameter Models
• This module equips students with a solid foundation in Bayesian inference for single parameter models, emphasizing both theoretical understanding and practical application.
• Module 3: Multiparameter Models
• This module provides an overview of Bayesian inference for multiparameter models, focusing on handling normal data, employing conjugate priors, and applying multivariate normal models to practical scenarios.
• Module 4: Large-Sample Inference and Frequency Properties
• This module provides an understanding of large-sample inference and frequency properties in Bayesian analysis, focusing on normal approximations, large-sample theory, and the evaluation of Bayesian methods from a frequentist perspective.
• Module 5: Hierarchical Models
• This module provides an overview of hierarchical models within Bayesian inference, focusing on constructing priors, understanding exchangeability, performing analysis, and ensuring model validity and improvement.
• Module 6: Bayesian Computation
• This module provides a comprehensive understanding of Bayesian computation techniques, emphasizing numerical integration, simulation methods, and advanced Markov chain algorithms. Students will gain practical skills in implementing these methods and debugging computational issues.
• Module 7: Regression Models
• This module consists of an overview of regression models in Bayesian inference, focusing on foundational principles, hierarchical linear models, and generalized linear models, with practical applications and advanced techniques.
• Module 8: Advanced Topics
• This module covers advanced topics in Bayesian inference, focusing on the setup, interpretation, and application of mixture models, as well as addressing computational challenges and integrating mixture models with multivariate data analysis.
• Summative Course Assessment
• This module contains the summative course assessment that has been designed to evaluate your understanding of the course material and assess your ability to apply the knowledge you have acquired throughout the course.