De Finetti's theorem

De Finetti's theorem states that an infinitely exchangeable sequence of random variables is a mixture of independent, identically distributed sequences, conditional on a latent parameter.

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Exchangeability: A joint distribution is exchangeable when it is unchanged by every finite permutation of its variables. The theorem begins with this permutation symmetry across the infinite sequence.
Bruno de Finetti: An Italian probabilist who developed a subjective interpretation of probability and proved a representation theorem for exchangeable sequences. His 1930 work established the result’s foundational form for binary sequences.
Infinite sequence: An ordered collection of random variables indexed by the positive integers. The representation requires exchangeability across an indefinitely extendable sequence.
Bayesian statistics: An approach to statistical inference that updates probability distributions for unknown quantities using observed data. The latent mixing law gives a principled prior representation for exchangeable observations.
Kolmogorov extension theorem: A theorem giving conditions under which consistent finite-dimensional distributions define a probability measure on an infinite product space. It constructs infinite sequences from finite laws, whereas de Finetti represents already exchangeable ones.
Bernoulli process: A sequence of independent, identically distributed binary random variables with a fixed success probability. For binary sequences, the mixing parameter is the latent success probability.
Richard von Mises: A mathematician who developed a frequency-based account of probability using infinite sequences of outcomes. His sequence-based approach formed part of the intellectual setting for de Finetti’s work.
Joint distribution: A probability distribution describing the values of two or more random variables together. Exchangeability is a symmetry property of the sequence’s joint distributions.
Beta-binomial distribution: A count distribution obtained by drawing a success probability from a beta distribution and then sampling Bernoulli trials. It is the finite-count predictive distribution from a beta mixture of Bernoulli sequences.
Aldous–Hoover representation theorem: A representation theorem expressing jointly exchangeable arrays as measurable functions of independent latent random variables. It extends representation ideas from sequences to exchangeable arrays.