Extensions
Versions of de Finetti's theorem for finitely exchangeable sequences, and for Markov exchangeable sequences have been proved by Diaconis and Freedman and by Kerns and Szekely. Two notions of partial exchangeability of arrays, known as separate and joint exchangeability lead to extensions of de Finetti's theorem for arrays by Aldous and Hoover.
The computable de Finetti theorem shows that if an exchangeable sequence of real random variables is given by a computer program, then a program which samples from the mixing measure can be automatically recovered.
In the setting of free probability, there is a noncommutative extension of de Finetti's theorem which characterizes noncommutative sequences invariant under quantum permutations.
See also: Choquet theory and Krein–Milman theoremRead more about this topic: De Finetti's Theorem
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