Overview

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This course equips learners with the theoretical knowledge and computational skills needed to implement modern Bayesian statistical methods in real-world settings. By completing the course, learners will be able to build and fit Bayesian models, apply computational algorithms for posterior inference, and interpret uncertainty in complex data analysis problems. Topics include maximum a posteriori (MAP) estimation, rejection sampling, and Markov chain Monte Carlo (MCMC) methods such as the Gibbs sampler and Metropolis-Hastings algorithms. Learners will also gain hands-on experience using Stan, one of the leading platforms for Bayesian modeling and probabilistic programming. The course is designed for learners seeking to strengthen their statistical, machine learning, and data science capabilities in industry or research settings. Bayesian methods are increasingly used in areas such as AI, forecasting, experimentation, risk analysis, and decision-making under uncertainty. Unlike many applied courses that focus primarily on software tools, this course emphasizes both the mathematical foundations and computational intuition underlying modern Bayesian workflows, helping learners develop a deeper understanding of how and why these methods work. This course can be taken for academic credit as part of CU Boulder’s Master of Science in Data Science (MS-DS) degree offered on the Coursera platform. The MS-DS is an interdisciplinary degree that brings together faculty from CU Boulder’s departments of Applied Mathematics, Computer Science, Information Science, and others. With performance-based admissions and no application process, the MS-DS is ideal for individuals with a broad range of undergraduate education and/or professional experience in computer science, information science, mathematics, and statistics. Learn more about the MS-DS program at https://www.coursera.org/degrees/master-of-science-data-science-boulder.

Syllabus

Introduction to Computational Bayesian Statistics

Some Bayesian inference problems are easily solved with basic algebra and calculus. For example, with a beta prior distribution over the probability of success in a binomial process, it is easy to show that the posterior distribution over the probability of success is also a beta distribution. However, many other, more complicated problems are not as easily solved. Instead, they require computational methods for approximating posterior distributions and their summary statistics. In this module, students will learn some computational algorithms for posterior distribution summaries, including the gradient ascent algorithm for calculating the MAP (maximum a posteriori) estimator, and Monte Carlo methods for computing other summary statistics from the posterior distribution.

Rejection Sampling

In this module, we introduce rejection sampling as a means of producing independent draws from a posterior density distribution where the density distribution's normalizing constant might not be known.

Gibbs Sampling

This module focuses on Gibbs sampling which is an Markov Chain Monte Carlo (MCMC) method for generating random draws from a posterior density distribution when the distribution of one model parameter conditioned on the other model parameters is known.

Metropolis Hastings Sampling

This module introduces the Metropolis sampling algorithm, another MCMC method for generating approximately independent, random draws from a posterior density distribution. The module also covers the Metropolis-Hastings extension of the Metropolis sampling algorithm and ends with a brief overview of some of the adaptations to the Metropolis-Hastings algorithm.

STAN

This module introduces STAN and demonstrates its use in R using Google Colab. STAN provides an efficient implementation of an adaptive Metropolis-Hastings algorithm, to overcome some of the limitations of the Metropolis-Hastings algorithm.