Exchangeable random variables: Difference between revisions

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is the same as the joint probability distribution of the original sequence.<ref name="ChowTeicher" >In short, the order of the sequence of random variables does not affect its joint probability distribution.

* Chow, Yuan Shih and Teicher, Henry, ''Probability theory. Independence, interchangeability, martingales,'' Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp.&nbsp;{{ISBN |0-387-98228-0}}</ref><ref>Aldous, David J., ''Exchangeability and related topics'', in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp.&nbsp;1–198, Springer, Berlin, 1985. {{ISBN |978-3-540-15203-3}} {{doi|10.1007/BFb0099421}}</ref>

(A sequence ''E''₁, ''E''₂, ''E''₃, ... of events is said to be exchangeable precisely if the sequence of its [[indicator function]]s is exchangeable.) The distribution function ''F''_{''X''_''1'',...,''X''_''n''}(''x''₁, ..., ''x''_''n'') of a finite sequence of exchangeable random variables is symmetric in its arguments {{nowrap|''x''₁, ... ,''x''_''n''.}} [[Olav Kallenberg]] provided an appropriate definition of exchangeability for continuous-time stochastic processes.<ref>{{cite journal | last=Diaconis| first=Persi | authorlink=Persi Diaconis | title=Book review: ''Probabilistic symmetries and invariance principles'' (Olav Kallenberg, Springer, New York, 2005) | journal=Bulletin of the Amererican Mathematical Society (New Series) | volume=46 | year=2009 | issue=4 | doi=10.1090/S0273-0979-09-01262-2 | url=http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01262-2/home.html | pages=691–696 | mr=2525743}}</ref><ref name="Kallenberg" >[[Olav Kallenberg|Kallenberg, O.]], ''Probabilistic symmetries and invariance principles''. Springer-Verlag, New York (2005). 510 pp.&nbsp;{{ISBN |0-387-25115-4}}.</ref>

== History ==

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== Examples ==

* Any [[convex combination]] or [[mixture distribution]] of [[iid]] sequences of random variables is exchangeable. A converse proposition is [[de Finetti's theorem]].<ref>Spizzichino, Fabio ''Subjective probability models for lifetimes''. Monographs on Statistics and Applied Probability, 91. ''Chapman & Hall/CRC'', Boca Raton, FL, 2001. xx+248 pp.&nbsp;{{ISBN |1-58488-060-0}}

</ref>

* Suppose an [[urn model|urn]] contains ''n'' red and ''m'' blue marbles. Suppose marbles are drawn without replacement until the urn is empty. Let ''X''_''i'' be the indicator random variable of the event that the ''i''th marble drawn is red. Then {''X''_''i''}_{''i''=1,...''n''} is an exchangeable sequence. This sequence cannot be extended to any longer exchangeable sequence.

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== Bibliography ==

* Aldous, David J., ''Exchangeability and related topics'', in: École d'Été de Probabilités de Saint-Flour XIII — 1983, Lecture Notes in Math. 1117, pp.&nbsp;1–198, Springer, Berlin, 1985. {{ISBN |978-3-540-15203-3}} {{doi|10.1007/BFb0099421}}

* Barlow, R. E. & Irony, T. Z. (1992) "Foundations of statistical quality control" in Ghosh, M. & Pathak, P.K. (eds.) ''Current Issues in Statistical Inference: Essays in Honor of D. Basu'', Hayward, CA: Institute of Mathematical Statistics, 99-112.

* Bergman, B. (2009) "Conceptualistic Pragmatism: A framework for Bayesian analysis?", ''IIE Transactions'', '''41''', 86–93

* {{cite book |last=Borovskikh | first=Yu. V. | title=''U''-statistics in Banach spaces | publisher=VSP | ___location=Utrecht | year=1996 | pages=xii+420 | isbn=90-6764-200-2 | mr=1419498 }}

* Chow, Yuan Shih and Teicher, Henry, ''Probability theory. Independence, interchangeability, martingales,'' Springer Texts in Statistics, 3rd ed., Springer, New York, 1997. xxii+488 pp.&nbsp;{{ISBN |0-387-98228-0}}

* {{cite journal | last=Diaconis| first=Persi | authorlink=Persi Diaconis | title=Book review: ''Probabilistic symmetries and invariance principles'' (Olav Kallenberg, Springer, New York, 2005) | journal=Bulletin of the Amererican Mathematical Society (New Series) | volume=46 | year=2009 | issue=4 | doi=10.1090/S0273-0979-09-01262-2 | url=http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01262-2/home.html | pages=691–696 | mr=2525743}}

* [[Olav Kallenberg|Kallenberg, O.]], ''Probabilistic symmetries and invariance principles''. Springer-Verlag, New York (2005). 510 pp.&nbsp;{{ISBN |0-387-25115-4}}.

* Kingman, J. F. C., ''Uses of exchangeability'', Ann. Probability 6 (1978) 83&ndash;197 {{MR|494344}} {{JSTOR|2243211}}

* O'Neill, B. (2009) Exchangeability, Correlation and Bayes' Effect. ''International Statistical Review'' '''77(2)''', pp.&nbsp;241–250. {{ISBN |978-3-540-15203-3}} {{doi|10.1111/j.1751-5823.2008.00059.x}}

* {{cite book|title=Limit theorems for sums of exchangeable random&nbsp;variables|first1=Robert Lee|last1=Taylor|first2=Peter Z.|last2=Daffer|first3=Ronald F.|last3=Patterson|mr=|ref=harv|publisher=Rowman and Allanheld|url=https://books.google.com/books?id=6RaoAAAAIAAJ|year=1985|pages=1–152}}

* Zabell, S. L. (1988) "Symmetry and its discontents", in Skyrms, B. & Harper, W. L. ''Causation, Chance and Credence, ''pp''155-190, Kluwer