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De Finetti's theorem
In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability 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".
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.
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.
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 natural number n and any finite sequence i1, ..., in and any permutation of the sequence π:{i1, ..., in } → {i1, ..., in },
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 Pólya urn model.
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De Finetti's theorem
In probability theory, de Finetti's theorem states that exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability 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".
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.
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.
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 natural number n and any finite sequence i1, ..., in and any permutation of the sequence π:{i1, ..., in } → {i1, ..., in },
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 Pólya urn model.