It is closely related to the use of independent and identically distributed random variables in statistical models.
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.
Olav Kallenberg provided an appropriate definition of exchangeability for continuous-time stochastic processes.
[5] Exchangeability is equivalent to the concept of statistical control introduced by Walter Shewhart also in 1924.
[6][7] The property of exchangeability is closely related to the use of independent and identically distributed (i.i.d.)
[8] A sequence of random variables that are i.i.d, conditional on some underlying distributional form, is exchangeable.
The converse can be established for infinite sequences, through an important representation theorem by Bruno de Finetti (later extended by other probability theorists such as Halmos and Savage).
[9] The extended versions of the theorem show that in any infinite sequence of exchangeable random variables, the random variables are conditionally independent and identically-distributed, given the underlying distributional form.
Another way of putting this is that de Finetti's theorem characterizes exchangeable sequences as mixtures of i.i.d.
sequences—while an exchangeable sequence need not itself be unconditionally i.i.d., it can be expressed as a mixture of underlying i.i.d.
random variables, based on some underlying distributional form.
An infinite exchangeable sequence is strictly stationary and so a law of large numbers in the form of Birkhoff–Khinchin theorem applies.
The close relationship between exchangeable sequences of random variables and the i.i.d.
This notion is central to Bruno de Finetti's development of predictive inference and to Bayesian statistics.
It can also be shown to be a useful foundational assumption in frequentist statistics and to link the two paradigms.
[10] The representation theorem: This statement is based on the presentation in O'Neill (2009) in references below.
This latter limit always exists for sums of indicator functions, so that the empirical distribution is always well-defined.)
There is a weaker lower bound than for infinite exchangeability and it is possible for negative correlation to exist.
Using the fact that the values are exchangeable, we have We can then solve the inequality for the covariance yielding the stated lower bound.
Exchangeable random variables arise in the study of U statistics, particularly in the Hoeffding decomposition.
[13] Exchangeability is a key assumption of the distribution-free inference method of conformal prediction.