Exchangeable Sequences

Abstract

Recall from Definition 4 of Chapter 22 that a finite or infinite sequence is exchangeable if its distribution is invariant under permutations of its terms. Our main goal in this chapter is to develop some simple ways to describe all exchangeable sequences. We will see that an infinite sequence is exchangeable if and only if it is conditionally iid, and that finite exchangeable sequences can be described in terms of a certain urn model. The characterization results, known as De Finetti Theorems, were foreshadowed in Example 2 and Problem 9 of Chapter 22. Problem 10 of Chapter 22 provides an example of a finite exchangeable sequence that is not conditionally iid, thereby showing that the finite case cannot be derived from the infinite case. We focus on exchangeable sequences whose terms take values in a finite set, and describe how some of the formulas obtained for such sequences can be used to obtain information about unknown parameters. Later we generalize to sequences whose terms take values in a Borel space. In the last section, we introduce some important distributions that are naturally connected with infinite exchangeable sequences and an urn model.