Exchangeable random variables: Difference between revisions

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== Exchangeability and the i.i.d. statistical model ==

The property of exchangeability is closely related to the use of [[independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) random variables in statistical models.<ref>{{cite book |last=Cordani |first=L. K. |last2=Wechsler |first2=S. |year=2006 |chapter=Teaching independence and exchangeability |title=Proceedings of the International Conference on Teaching Statistics |___location=Den Haag |yearpublisher=2006International Association for Statistical Education |chapter-url=https://iase-web.org/documents/papers/icots7/3I1_CORD.pdf?1402524964 }}</ref> A sequence of random variables that are 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.

Mixtures of exchangeable sequences (in particular, sequences of i.i.d. variables) are exchangeable. The converse can be established for infinite sequences, through an important [[de Finetti's theorem|representation theorem]] by [[Bruno de Finetti]] (later extended by other probability theorists such as [[Paul Halmos|Halmos]] and [[Leonard Jimmie Savage|Savage]]).<ref>{{cite book |first=P. |last=Diaconis |authorlink=Persi Diaconis |chapter=Recent Progress on de Finetti's Notions of Exchangeability |pages=111–125 |year=1988 |title=Bayesian Statistics |volume=3 |editor1-first=J. M. |editor1-last=Bernardo |editor1-link=José-Miguel Bernardo |editor2-first=M. H. |editor2-last=DeGroot |editor3-first=D. V. |editor3-last=Lindley |editor4-first=A. F. M. |editor4-last=Smith |display-editors=1 |publisher=Oxford University Press |isbn=0-19-852220-7 }}</ref> 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 random variables|independent and identically-distributed]], given the underlying distributional form. This theorem is stated briefly below. (De Finetti's original theorem only showed this to be true for random indicator variables, but this was later extended to encompass all sequences of random variables.) 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. sequences.<ref name="ChowTeicher"/>