Hierarchical Models and Gibbs Sampling

Understanding Computational Bayesian Statistics. By William M. Bolstad Copyright 2010 John Wiley Sons. Inc. 

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In the previous example, we found the exact marginal posterior distribution for each of the individual means analytically. They were the parameters of interest and the hyperparameter was considered a nuisance parameter. In many cases we cannot find the posterior analytically. Instead we use the Gibbs sampler to draw a sample from the joint posterior of all the parameters for the hierarchical model. The Gibbs sample for a particular parameter is a sample from its marginal posterior. We will base our inference about that parameter on the thinned sample from its marginal posterior. We don t even have to marginalize out the other parameters as looking at the sample for that particular parameter does it automatically. 

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