Most applications of MCMC to date, including the majority of those described in the following chapters, are oriented towards Bayesian inference. From a Bayesian perspective, there is no fundamental distinction between

The integrations in this expression have until recently been the source of most of the practical difficulties in Bayesian inference, especially in high dimensions. In most applications, analytic evaluation of E[J(0)ID] is impossible. Alternative approaches include numerical evaluation, which is difficult and inaccurate in greater than about 20 dimensions; analytic approximation such as the Laplace approximation (Kass et al., 1988), which is sometimes appropriate; and Monte Carlo integration, including MCMC.