Exchangeable

Exchangeable is an adjective used to describe items, variables, or processes that can be exchanged without affecting the outcome or interpretation. In everyday use, two goods are exchangeable if their identities do not matter for a given transaction or comparison. In science and mathematics, the term has a more precise technical meaning.

In probability theory, a sequence of random variables X1, X2, ... is called exchangeable if its joint distribution is invariant under finite permutations of the indices. In other words, for any n and any permutation pi of {1,...,n}, the distribution of (X1,...,Xn) is the same as that of (X_pi(1),...,X_pi(n)). Exchangeability is weaker than independence: exchangeable variables may be dependent, but they share the same marginal distributions and your model does not privilege any order.

An important result is de Finetti's theorem: any infinite exchangeable sequence is conditionally independent and identically distributed given a latent random element, often interpreted as a random measure or parameter. This means an observed exchangeable sequence can be viewed as a mixture of i.i.d. sequences. Classic examples include draws from an urn with unknown composition, where only the mixture parameter governs all draws, and, for finite populations, several orderings lead to exchangeable joint distributions.

Usage and implications: assuming exchangeability is common in Bayesian statistics and probabilistic modeling because it justifies pooling information across observations while allowing dependence. It clarifies when models fit data without assuming a specific order and underpins certain nonparametric approaches.