De Finetti's Theorem

De Finetti's Theorem

In probability theory, de Finetti's theorem explains why exchangeable observations are conditionally independent given some latent variable to which an epistemic probability distribution would then be assigned. It is named in honor of Bruno de Finetti.

It states that an exchangeable sequence of Bernoulli random variables is a "mixture" of independent and identically distributed (i.i.d.) Bernoulli random variables – while the individual variables of the exchangeable sequence are not themselves i.i.d., only exchangeable, there is an underlying family of i.i.d. random variables.

Thus, while observations need not be i.i.d. for a sequence to be exchangeable, there are underlying, generally unobservable, quantities which are i.i.d. – exchangeable sequences are (not necessarily i.i.d.) mixtures of i.i.d. sequences.