Editing De Finetti's theorem (section)

== Statement of the theorem ==

A [[random variable]] ''X'' has a [[Bernoulli distribution]] if Pr(''X''&nbsp;=&nbsp;1)&nbsp;=&nbsp;''p'' and Pr(''X''&nbsp;=&nbsp;0)&nbsp;=&nbsp;1&nbsp;&minus;&nbsp;''p'' for some ''p''&nbsp;∈&nbsp;(0,&nbsp;1).

De Finetti's theorem states that the probability distribution of any infinite exchangeable sequence of Bernoulli random variables is a "[[mixture distribution|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 [[compound probability distribution|integral over a measure]]  rather than a sum.

More precisely, suppose ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... is an infinite exchangeable sequence of Bernoulli-distributed random variables.  Then there is some probability measure ''m'' on the interval [0,&nbsp;1] and some random variable ''Y'' such that
* The probability measure of ''Y'' is ''m'', and
* The [[conditional probability distribution]] of the whole sequence ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... given the value of ''Y'' is described by saying that
** ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ... are [[conditional independence|conditionally independent]] given ''Y'', and
** For any ''i'' ∈ {1, 2, 3, ...}, the conditional probability that ''X''<sub>''i''</sub> = 1, given the value of ''Y'', is ''Y''.

=== Another way of stating the theorem ===

Suppose <math>X_1,X_2,X_3,\ldots</math> is an infinite exchangeable sequence of Bernoulli random variables.  Then <math>X_1,X_2,X_3,\ldots</math> are conditionally independent and identically distributed given the [[Invariant sigma-algebra#Exchangeable sigma-algebra|exchangeable sigma-algebra]] (i.e., the sigma-algebra consisting of events that are measurable with respect to <math>X_1,X_2,\ldots</math> and invariant under finite permutations of the indices).

=== A plain-language consequence of the theorem ===
According to [[David Spiegelhalter]] (ref 1) the theorem provides a pragmatic approach to de Finetti's statement that "Probability does not exist". If our view of the probability of a sequence of events is subjective but remains unaffected by the order in which we make our observations, then the sequence can be regarded as [[Exchangeable_random_variables|''exchangeable'']]. De Finetti's theorem then implies that believing the sequence to be exchangeable is mathematically equivalent to acting as if the events are ''independent'' and have an objective underlying probability of occurring, with our uncertainty about what that probability is being expressed by a subjective probability distribution function. According to Spiegelhalter:  ″This is remarkable: it shows that, starting from a specific, but purely subjective, expression of convictions, we should act as if events were driven by objective chances."