Summary. This chapter deals in outline with a number of topics that fall outside the main theme of the book. The topics are prediction, decision analysis and point estimation, concentrating especially on estimates that are exactly or approximately unbiased. Finally some isolated remarks are made about methods, especially for relatively complicated models, that avoid direct use of the likelihood.

Prediction

In prediction problems the target of study is not a parameter but the value of an unobserved random variable. This includes, however, in so-called hierarchical models estimating the value of a random parameter attached to a particular portion of the data. In Bayesian theory the formal distinction between prediction and estimation largely disappears in that all unknowns have probability distributions. In frequentist theory the simplest approach is to use Bayes' theorem to find the distribution of the aspect of interest and to replace unknown parameters by good estimates. In special cases more refined treatment is possible.

In the special case when the value Y*, say, to be predicted is conditionally independent of the data given the parameters the Bayesian solution is particularly simple. A predictive distribution is found by averaging the density fY* (y*; θ) over the posterior distribution of the parameter.

In special cases a formally exact frequentist predictive distribution is obtained by the following device.