Part III: Advanced Computation

The remainder of this book delves into more sophisticated models. Before we begin this enterprise, however, we detour to describe methods for computing posterior distributions in hierarchical models. Toward the end of Chapter 5, the algebra required for analytic derivation of posterior distributions became less and less attractive, and that was with a relatively simple model constructed entirely from normal distributions. If we try to solve more complicated problems analytically, the algebra starts to overwhelm the statistical science almost entirely, making the full Bayesian analysis of realistic probability models too cumbersome for most practical applications. Fortunately, a battery of powerful methods has been developed over the past few decades for approximating and simulating from probability distributions. In the next four chapters, we survey some useful simulation methods that we apply in later chapters in the context of specific models. Some of the simpler simulation methods we present here have already been introduced in examples in earlier chapters.

Because the focus of this book is on data analysis rather than computation, we move through the material of Part III briskly, with the intent that it be used as a reference when applying the models discussed in Parts IV and V. We have also attempted to place a variety of useful techniques in the context of a systematic general approach to Bayesian computation. Our general philosophy in computation, as in modeling, is pluralistic, developing approximations using a variety of techniques.