Statement of The Theorem
A random variable X has a Bernoulli distribution if Pr(X = 1) = p and Pr(X = 0) = 1 − p for some p ∈ (0, 1).
De Finetti's theorem states that the probability distribution of any infinite exchangeable sequence of Bernoulli random variables is a "mixture" of the probability distributions of independent and identically distributed sequences of Bernoulli random variables. "Mixture", in this sense, means a weighted average, but this need not mean a finite or countably infinite (i.e., discrete) weighted average: it can be an integral rather than a sum.
More precisely, suppose X1, X2, X3, ... is an infinite exchangeable sequence of Bernoulli-distributed random variables. Then there is some probability distribution m on the interval and some random variable Y such that
- The probability distribution of Y is m, and
- The conditional probability distribution of the whole sequence X1, X2, X3, ... given the value of Y is described by saying that
- X1, X2, X3, ... are conditionally independent given Y, and
- For any i ∈ {1, 2, 3, ...}, the conditional probability that Xi = 1, given the value of Y, is Y.
Read more about this topic: De Finetti's Theorem
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