A Tutorial on Markov Chain Monte-Carlo and Bayesian Modeling

Abstract

This tutorial provides an introduction to Bayesian modeling and Markov Chain Monte-Carlo (MCMC) algorithms including the Metropolis-Hastings and Gibbs Sampling algorithms. We discuss some of the challenges associated with running MCMC including the important question of determining when convergence to stationarity has been achieved. Several applications of Bayesian modeling are also provided including the MRP approach to modeling election outcomes, topic modeling in machine learning, and large-scale optimization and code breaking. We also discuss the important problems of Bayesian model checking and selection and provide an introduction to empirical Bayesian models.

In the appendices we briefly discuss several topics including: (i) Hamiltonian Monte-Carlo (an MCMC algorithm that forms the basis for the statistical modeling language STAN) (ii) the EM algorithm and (iii) other problems in Bayesian statistics including Bayesian approaches to constructing confidence intervals, hypothesis testing and decision theory.