Introduction to Bayesian Computation

In the previous two chapters, two types of strategies were used in the summarization of posterior distributions. If the sampling density has a familiar functional form, such as a member of an exponential family, and a conjugate prior is chosen for the parameter, then the posterior distribution often is expressible in terms of familiar probability distributions. In this case, we can simulate parameters directly by use of the R collection of random variate functions (such as rnorm, rbeta and rgamma), and we can summarize the posterior by computations on this simulated sample. A second type of computing strategy is what we called the “brute-force” method. In the case where the posterior distribution is not a familiar functional form, then one simply computes values of the posterior on a grid of points and then approximates the continuous posterior by a discrete posterior that is concentrated on the values of the grid. This brute-force method can be generally applied for oneand two-parameter problems such as those illustrated in Chapters 3 and 4.