De Finetti's theorem - Wikipedia

Abstraction: Exchangeable observations are conditionally independent given latent variable

Key points:

• States that an infinite exchangeable sequence of random variables is a mixture of i.i.d. sequences, with the mixing measure on [0,1]
• Exchangeability (joint distribution unchanged by permutation of indices) implies conditional independence relative to a latent variable, but not unconditional independence
• For Bernoulli sequences, P(sequence) can be written as an integral over Bernoulli(p) measures weighted by a probability measure on p in [0,1]
• Provides mathematical justification for Bayesian reasoning: subjective exchangeability is equivalent to acting as if events have objective probabilities
• Theorem is false in general for finite exchangeable sequences; requires infinite exchangeability or extendability to infinite sequences
• Extensions exist for Markov-exchangeable sequences (Diaconis-Freedman), arrays (Aldous-Hoover), free probability (noncommutative), and quantum states

Connections: Bruno De Finetti · David Spiegelhalter · Exchangeability · Conditional Independence · Bayesian Inference · Mixture Distributions

Source: http://en.wikipedia.org/wiki/De_Finetti's_theorem