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# Commentary: Note on exchangeability and de Finetti's Theorem

In several places Jaynes refers to exchangeability and de Finetti's Theorem without defining these; finally, in Chapter 18 section 16 (p. 586 onward) he says a little bit about just what de Finetti's Theorem is. For those readers who are unfamiliar with this topic, Bernardo and Smith's book Bayesian Theory has a nice discussion in sections 4.2 and 4.3. Here is a brief, simplified summary of the definitions and theorems given therein, translated into the vocabulary and notation of Jaynes.

Notation. We write for , and for the infinite sequence . For simplicity, we assume that the set of possible values for each variable is the same finite set .

Definition of finite exchangeability. The variables are said to be (finitely) exchangeable for a state of information if, for any constant values , we have

whenever is a permutation of the variables .

Definition of infinite exchangeability. The infinite sequence of variables is said to be infinitely exchangeable for a state of information if every finite subsequence of is exchangeable for .

Theorem. Let be the set of probability mass functions over . If is an infinitely exchangeable sequence of variables for , then there exists a probability density over such that

for any constant . (Note that may involve delta functions, to assign positive probability to a single specific value in .)

In other words, we may reason as if there exists some additional variable such that the variables are independently and identically distributed when the value of is known, and is .