ABSTRACT
As discussed in Chapter 2, determination of posterior distributions comes down to the evaluation of complex, often high-dimensional integrals (i.e., the denominator of expression (2.1)). In addition, posterior summarization often involves computing moments or quantiles, which leads to more integration (i.e., now integrating the numerator of expression (2.1)). A solution to an important special case of the problem was provided in Subsection 2.2.2, which described how conjugate prior forms may often be found that enable at least partial analytic evaluation of these integrals. Still, in all but the simplest model settings (typically linear models with normal likelihoods), some intractable integrations remain. In the last chapter, we were content to let WinBUGS handle implementation of the necessary computational methods. In this chapter, we present a full description of the most commonly used such methods, for readers whose MCMC needs may exceed the capabilities of WinBUGS and other standard software packages.
