Definition of Term: Exchangeability

Exchangeability is regarded as a fundamental concept in any statistical modelling – regardless of whether that analysis is performed from a Bayesian or frequentist perspective. A sequence of random variables is described as exchangeable if our beliefs about that collection are unaffected by the order in which they appear. For example, if we perform 100 tosses of a coin, then, if the coin tosses are exchangeable, our beliefs about the fairness of the coin will be unaffected by the order in which we observe the heads and tails.

In formal terms, when our beliefs about a sequence of random variables \(X_1, X_2, \ldots\) are characterised by a joint probability distribution, if the \(X_i\)s are exchangeable then the joint probability distribution of the sequence is the same as the joint distribution over any re-ordering of the \(X_i\)s. In fact, the assertion of exchangeability implies the existence of this underlying joint distribution.

This idea of exchangeability underpins the concept of prediction, as when past observations and future observations are exchangeable then the future will be predictable on the basis of the data gathered in the past.