A 2-Day Seminar Taught by Scott Lynch, Ph.D.

This seminar provides a detailed introduction to applied Bayesian statistics. The course begins with a brief review of the “classical” statistical approach involving maximum likelihood methods. I then develop the Bayesian approach. First, Bayes’ Theorem is introduced with a hypothetical example, which is extended to illustrate Bayes’ Theorem for probability distributions, including the concepts of “prior” and “posterior distributions.”

The seminar will then present a few simple examples using real data, contrasting the classical approach with the Bayesian approach and showing how results are presented and interpreted in the Bayesian paradigm using summaries of the posterior distribution. Finally, the seminar will show the derivation of posterior distributions for some standard, basic models in order to illustrate the algebra and calculus required to complete Bayesian analyses.

Following this theoretical introduction, the seminar will turn to developing and explaining sampling methods. Unlike likelihood analysis, which involves differential calculus to obtain estimates, Bayesian analysis involves integral calculus to obtain summaries of the posterior distribution. Integration can be performed analytically or computationally, but high-dimensional integration is difficult using either approach.

Sampling methods provide an alternative approach to computing integrals, and it is the development of Markov chain Monte Carlo (MCMC) sampling methods—coupled with the rapid increase in computing power in the 1990s—that led to the explosion in the use of Bayesian methods. Most contemporary Bayesian analyses use MCMC or similar sampling methods, and the course will spend considerable time explaining, demonstrating, and applying MCMC methods in R to several common models in social science research, including the linear model, generalized linear models, and hierarchical models.

Finally, the seminar will show how posterior predictive simulation can be used to evaluate model fit—both to the overall sample and to individual cases—but perhaps more importantly, to produce distributions of ancillary measures that are functions of model parameters. For example, life expectancies are an important and easily understood measure of population health and may be computed from hazard model parameters. Thus, we may sample parameters from the posterior distributions for a hazard model using MCMC methods and then compute life tables from these parameter samples applied to specific covariate values/profiles to obtain distributions of life expectancies for specific subpopulations.

By the end of the seminar, one should understand the basic concepts underlying the Bayesian approach and be able to conduct basic Bayesian analyses in R for common models used in social science research.