Exchangeability and de Finetti's Theorem

Exchangeability formalizes the idea that the order of observations carries no information, and de Finetti's theorem shows this assumption justifies treating data as conditionally i.i.d. given a parameter with a prior.

Definition

A sequence of random variables is exchangeable if its joint distribution is invariant under any permutation of indices; de Finetti's theorem states that an infinite exchangeable sequence is a mixture of i.i.d. sequences, with the mixing distribution playing the role of a prior.

Scope

This topic covers finite and infinite exchangeability, de Finetti's representation theorem and its role in grounding parametric models and priors on purely subjective probability, and partial exchangeability for structured data.

Core questions

• What does it mean for a sequence of observations to be exchangeable?
• How does de Finetti's theorem represent an exchangeable sequence as conditionally i.i.d.?
• Why does exchangeability provide a subjective-probability justification for priors and parametric models?
• How is the idea extended through partial exchangeability to structured or grouped data?

Key concepts

• exchangeability
• permutation invariance
• mixing distribution
• conditional independence
• partial exchangeability
• subjective probability

Key theories

De Finetti's representation theorem

Any infinite exchangeable binary sequence can be written as a mixture of Bernoulli sequences, with the mixing measure interpretable as a prior over the success probability; the result generalizes to broader observation spaces.

Partial exchangeability

When data fall into groups, exchangeability is assumed within groups, motivating hierarchical models in which group-level parameters are themselves exchangeable.

Clinical relevance

Exchangeability is the modeling assumption that licenses pooling information across similar units, underlying meta-analysis, multi-center trials, and hierarchical models throughout the applied sciences.

History

De Finetti introduced exchangeability and proved his representation theorem in the 1930s, providing a subjective-probability alternative to the frequentist notion of i.i.d. sampling. Hewitt and Savage later extended the theorem to general spaces.

Key figures

• Bruno de Finetti
• David Hewitt
• Leonard J. Savage

Seminal works

• definetti1937
• bernardo1994

Frequently asked questions

Is exchangeability the same as independence?

No. Exchangeable variables are generally dependent, but de Finetti's theorem shows they become conditionally independent and identically distributed once an unknown parameter is introduced, which is exactly the structure of a Bayesian model.