De Finetti's theorem
In probability theory, de Finetti's theorem states that 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.
For the special case of an exchangeable sequence of Bernoulli random variables it states that such a sequence is a "mixture" of sequences 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.
A Bayesian statistician often seeks the conditional probability distribution of a random quantity given the data. The concept of exchangeability was introduced by de Finetti. De Finetti's theorem explains a mathematical relationship between independence and exchangeability.
An infinite sequence
of random variables is said to be exchangeable if for any finite cardinal number n and any two finite sequences i1, ..., in and j1, ..., jn (with each of the is distinct, and each of the js distinct), the two sequences
both have the same joint probability distribution.
If an identically distributed sequence is independent, then the sequence is exchangeable; however, the converse is false --- there exist exchangeable random variables that are not statistically independent, for example the Polya urn model.
...
Wikipedia