Editing De Finetti's theorem (section)

{{Short description|Conditional independence of exchangeable observations}}
In [[probability theory]], '''de Finetti's theorem''' states that [[exchangeable random variables|exchangeable]] observations are [[conditionally independent]] relative to some [[latent variable]]. An  [[epistemic probability]] [[probability distribution|distribution]] could then be assigned to this variable. It is named in honor of [[Bruno de Finetti]], and one of its uses is in providing a pragmatic approach to de Finetti's well-known dictum "Probability does not exist".<ref>Spiegelhalter, D. 2024 [https://www.scientificamerican.com/article/why-probability-probably-doesnt-exist-but-its-useful-to-act-like-it-does] Why probability doesn't exist (but it's useful to act like it does) ''Scientific American'' 26 December </ref>

For the special case of an exchangeable sequence of [[Bernoulli distribution|Bernoulli]] random variables it states that such a sequence is a "[[mixture distribution|mixture]]" of sequences of [[independent and identically distributed]] (i.i.d.) Bernoulli random variables.

A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of a finite set of indices. In general, while the variables of the exchangeable sequence are not ''themselves'' independent, only exchangeable, there is an ''underlying'' family of i.i.d. random variables. That is, there are underlying, generally unobservable, quantities that are i.i.d. – exchangeable sequences are mixtures of i.i.d. sequences.