Bayesian Statistics: An Introduction, 4th Edition

7.8 Empirical Bayes methods

7.8.1 Von Mises’ example

Only a very brief idea about empirical Bayes methods will be given in this chapter; more will be said about this topic in Chapter 8 and a full account can be found in Maritz and Lwin (1989). One of the reasons for this brief treatment is that, despite their name, very few empirical Bayes procedures are, in fact, Bayesian; for a discussion of this point see, for example, Deely and Lindley (1981).

The problems we will consider in this section are concerned with a sequence xi of observations such that the distribution of the ith observation xi depends on a parameter , typically in such a way that has the same functional form for all i. The parameters are themselves supposed to be a random sample from some (unknown) distribution, and it is this unknown distribution that plays the role of a prior distribution and so accounts for the use of the name of Bayes. There is a clear contrast with the situation in the rest of the book, where the prior distribution represents our prior beliefs, and so by definition it cannot be unknown. Further, the prior distribution in empirical Bayes methods is usually given a frequency interpretation, by contrast ...