Exchangeability and de Finetti's Theorem

Assuming a sequence of returns or trades is i.i.d. — independent and identically distributed — is convenient but often too strong: real observations are usually correlated through some shared underlying factor (a common regime, a shared risk driver), even when there's no reason to think any specific ordering of them matters. Exchangeability captures exactly that weaker assumption: a sequence $X_1, X_2, \dots, X_n$ is exchangeable if its joint distribution is unchanged by any reshuffling of the indices — $P(X_1, \dots, X_n) = P(X_{\pi(1)}, \dots, X_{\pi(n)})$ for any permutation $\pi$ — without requiring the variables to be independent of one another.

de Finetti's theorem says something remarkable about infinite exchangeable sequences: they behave exactly as if there's a hidden random parameter $\theta$, drawn once, such that conditional on $\theta$ the observations are i.i.d.:

$$P(X_1, \dots, X_n) = \int \prod_{i=1}^n P(X_i \mid \theta) \, dP(\theta) .$$

In plain English: exchangeable data looks like i.i.d. data generated from an unknown, fixed common cause — which is exactly the Bayesian picture of a prior over a parameter followed by i.i.d. draws given that parameter, rather than an assumption imposed by hand.

Worked example

A trader's daily win/loss outcomes across a year are not literally independent — a shared unknown "true edge" $\theta$ (perhaps 55% versus 52% win rate) links every day together, since knowing today was a win makes tomorrow slightly more likely to be a win too, purely because it updates your belief about $\theta$. But the days are still exchangeable: reordering them doesn't change the joint probability of the sequence of outcomes, since nothing about the process depends on calendar order. De Finetti's theorem says this is mathematically equivalent to: draw $\theta$ once from some prior, then flip a $\theta$-weighted coin independently each day — exactly the Bayesian model a quant would already reach for.