Exchangeable random variables: Difference between revisions

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AnIn [[statistics]], an '''exchangeable sequence of random variables''' (also sometimes '''interchangeable''') is a sequence such that future samples behave like earlier samples, meaning formally that any order (of a finite number of samples) is equally likely. This formalizes the notion of "the future being predictable on the basis of past experience."

sequence ''X''₁, ''X''₂, ''X''₃, ... of [[random variable]]s such that for any finite [[permutation]] σ of the indices 1, 2, 3, ..., i.e. any permutation σ that leaves all but finitely many indices fixed, the [[joint probability distribution]] of the permuted sequence

A sequence of [[independent and identically-distributed random variables]] (i.i.d.) is exchangeable, but so is [[simple random sample without replacement|sampling without replacement]], which is not independent.

The notion is central to [[Bruno de Finetti|Bruno de Finetti's]] development of [[predictive inference]] and to [[Bayesian statistics]] – where [[frequentist statistics]] uses i.i.d. variables (samples from a population), Bayesian statistics more frequently uses exchangeable sequences.

== Definition ==

Formally, an '''exchangeable sequence of random variables'''

is sequence ''X''₁, ''X''₂, ''X''₃, ... of [[random variable]]s such that for any finite [[permutation]] σ of the indices 1, 2, 3, ..., i.e. any permutation σ that leaves all but finitely many indices fixed, the [[joint probability distribution]] of the permuted sequence

:<math> X_{\sigma(1)}, X_{\sigma(2)}, X_{\sigma(3)}, \dots</math>