Exchangeability and de Finetti's Theorem

A sequence of random variables is exchangeable if any reordering of them has exactly the same joint probability distribution as the original order — for instance, the outcomes of repeated coin flips from the same coin are exchangeable, because "heads, tails, heads" and "tails, heads, heads" are equally likely combinations, even though the flips aren't assumed independent of each other in advance. Exchangeability is a weaker, more general condition than assuming the variables are independent and identically distributed (iid): iid implies exchangeable, but exchangeable does not require independence.

De Finetti's theorem says that any infinite exchangeable sequence of binary (or more general) random variables can be represented as if it were generated in two steps: first, a parameter — like a coin's true bias $\theta$ — is drawn once from some prior distribution, and then, conditional on that fixed $\theta$, all the individual outcomes are drawn independently. In other words, exchangeability alone, without any explicit prior assumption, forces the existence of an underlying parameter and a prior distribution over it that reproduces the observed dependence structure.

This is a foundational result for Bayesian statistics: it means treating repeated observations as exchangeable — a much weaker and more defensible assumption than asserting iid with a known model — already implies the Bayesian machinery of a prior over a parameter and independent draws conditional on it. A quant modeling a sequence of daily win/loss trade outcomes as exchangeable (reasonable if the underlying strategy's edge is stable but unknown) is implicitly committing to exactly this parameter-plus-independent-draws structure, whether or not a specific prior was ever written down.