Exchangeability
In 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." It is closely related to the use of independent and identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling.
Formally, an exchangeable sequence of random variables is a finite or infinite sequence X1, X2, X3, ... of random variables such that for any finite permutation σ of the indices 1, 2, 3, ..., (the permutation acts on only finitely many indices, with the rest fixed), the joint probability distribution of the permuted sequence
is the same as the joint probability distribution of the original sequence.
(A sequence E1, E2, E3, ... of events is said to be exchangeable precisely if the sequence of its indicator functions is exchangeable.) The distribution function FX1,...,Xn(x1, ..., xn) of a finite sequence of exchangeable random variables is symmetric in its arguments x1, ... ,xn. Olav Kallenberg provided an appropriate definition of exchangeability for continuous-time stochastic processes.
The concept was introduced by William Ernest Johnson in his 1924 book Logic, Part III: The Logical Foundations of Science. Exchangeability is equivalent to the concept of statistical control introduced by Walter Shewhart also in 1924.
The property of exchangeability is closely related to the use of independent and identically-distributed random variables in statistical models. A sequence of random variables that are independent and identically-distributed (i.i.d), conditional on some underlying distributional form is exchangeable. This follows directly from the structure of the joint probability distribution generated by the i.i.d form.
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