Bayesian Statistics and Marketing, 2nd Edition

3MCMC Methods

Abstract

This chapter provides an introduction to Markov Chain Monte Carlo (MCMC) methods and provides detailed discussion of the MCMC methods that have proved especially useful for marketing and micro‐econometric problems. In addition, several key examples are developed which help set the stage for more complicated models covered in later chapters. These include the Gibbs samplers for binary probit, mixture of normals and hierarchical linear models. In addition, Metropolis methods are introduced and illustrated with the multinomial logit model. Those readers who desire an introduction to MCMC methods without much theoretical background should skip section 3.3 on Markov Chain theory and only skim the beginning of 3.9 on Metropolis methods (skip over the proof following the introduction of the “Continuous State Space Metropolis” in section 3.10).

Given a model (prior and likelihood) or set of models, the computational phase of Bayesian inference requires practical methods for summarizing/exploring the posterior distribution. In many cases, the posterior distribution is represented by an un‐normalized density, , and the problem is to construct simulation‐based estimates of various aspects of this distribution. For any problem outside the conjugate family, the posterior density will be of a form for which analytical results on marginals or moments will be unavailable. ...