Sometimes it is easy to compute the posterior distribution analytically, but very often this is not possible. One can then use computer simulation to draw a sample of η-values from the posterior distribution. The mean or median of the posterior distribution can then be approximated by the mean or median of the sampled η-values, and approximate credibility regions can be determined in a similar way. In practice, the joint posterior distribution of all model parameters is usually quite complicated. Therefore, the complete set of parameters is split up into a number of subsets in such a way that the conditional posterior distribution of each subset given all other parameters has a tractable form and can be easily sampled from. This approach is known as Gibbs sampling (Geman and Geman 1984; Gelfand et al. 1990; Gelman et al. 2004), which is a special case of an MCMC algorithm. When however the conditional posterior distribution of a subset of the parameters is not easy or even impossible to sample from directly, other MCMC algorithms can be used, where one samples from a
