Optimising and Adapting the Metropolis Algorithm

Now, if π has simple form, then we might be able to compute such quantities analytically using elementary calculus. Or, if the dimension d is fairly small, then we might be able to use standard numerical integration techniques. But such simple approaches fail in many situations. For example, the variance components model is a typical statistical model which has been used to study such diverse topics as melanoma recurrence rates (Bartolucci et al., 2007), students’ success at law school (Rubin, 1980), comparison of baseball homerun hitters (Albert, 1992), analysis of fabric cyes (Box and Tiao, 1973), and more. The corresponding Bayesian posterior distribution for this model is given by: