Editing De Finetti's theorem (section)

== Extensions ==
Versions of de&nbsp;Finetti's theorem for  ''finite'' exchangeable sequences,<ref>{{cite journal
 |first1=P. |last1=Diaconis |author-link=Persi Diaconis
 |first2=D. |last2=Freedman |author-link2=David A. Freedman (statistician)
 |title=Finite exchangeable sequences
 |journal=Annals of Probability
 |volume=8 |issue=4 |year=1980 |pages=745&ndash;764
 |doi=10.1214/aop/1176994663 |mr=577313 | zbl = 0434.60034
|doi-access=free }}</ref><ref>{{cite journal
 |first1=G.&nbsp;J. |last1=Szekely |author-link=Gabor J Szekely
 |first2= J.&nbsp;G.  |last2=Kerns  |title=De&nbsp;Finetti's theorem for abstract finite exchangeable sequences |journal=Journal of Theoretical Probability |volume=19 |issue=3 |year=2006 |pages= 589–608 |doi=10.1007/s10959-006-0028-z|s2cid=119981020 }}
</ref> and for ''Markov&nbsp;exchangeable'' sequences<ref>{{cite journal
 |first1=P. |last1=Diaconis |author-link=Persi Diaconis
 |first2=D. |last2=Freedman |author-link2=David A. Freedman (statistician)
 |title=De&nbsp;Finetti's theorem for Markov chains
 |journal=Annals of Probability
 |volume=8 |issue=1 |year=1980 |pages=115&ndash;130
 |doi=10.1214/aop/1176994828 |mr=556418| zbl=0426.60064
|doi-access=free }}</ref> have been proved by Diaconis and Freedman and by Kerns and Szekely. 
Two notions of partial exchangeability of arrays, known as ''separate'' and ''joint exchangeability'' lead to extensions of de&nbsp;Finetti's theorem for arrays by Aldous and Hoover.<ref>Persi Diaconis and [[Svante Janson]] (2008) [http://www.stat.berkeley.edu/~aldous/Research/persi-svante.pdf "Graph Limits and Exchangeable Random Graphs"],''Rendiconti di&nbsp;Matematica'', Ser. VII 28(1), 33–61.</ref>

The computable de&nbsp;Finetti theorem shows that if an exchangeable sequence of real random variables is given by a computer program, then a program which samples from the mixing measure can be automatically recovered.<ref>
Cameron Freer and Daniel Roy (2009) [https://doi.org/10.1007%2F978-3-642-03073-4_23 "Computable exchangeable sequences have computable de&nbsp;Finetti measures"],  ''Proceedings of the 5th Conference on Computability in Europe: Mathematical Theory and Computational Practice'', Lecture Notes in Computer Science, Vol. 5635, pp. 218&ndash;231.</ref>

In the setting of [[free probability]], there is a noncommutative extension of de Finetti's theorem which characterizes noncommutative sequences invariant under quantum permutations.<ref>
{{cite journal |first1=Claus |last1=Koestler |first2=Roland |last2=Speicher |year=2009 |title=A noncommutative de&nbsp;Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation |journal=Commun. Math. Phys. |volume=291 |issue= 2|pages=473&ndash;490 |doi= 10.1007/s00220-009-0802-8|bibcode=2009CMaPh.291..473K |arxiv=0807.0677 |s2cid=115155584 }}
</ref>

Extensions of de Finetti's theorem to quantum states have been found to be useful in [[quantum information]],<ref>{{Cite journal|last1=Caves|first1=Carlton M.|last2=Fuchs|first2=Christopher A.|last3=Schack|first3=Ruediger|date=2002-08-20|title=Unknown quantum states: The quantum de Finetti representation|journal=Journal of Mathematical Physics|volume=43|issue=9|pages=4537–4559|arxiv=quant-ph/0104088|doi=10.1063/1.1494475|issn=0022-2488|bibcode=2002JMP....43.4537C|s2cid=17416262}}</ref><ref>{{cite web|url=http://math.ucr.edu/home/baez/week251.html|title=This Week's Finds in Mathematical Physics (Week 251)|year=2007|author=J. Baez|access-date=29 April 2012|author-link=John C. Baez}}</ref><ref>{{Cite book|last1=Brandao|first1=Fernando G.S.L.|last2=Harrow|first2=Aram W.|title=Proceedings of the forty-fifth annual ACM symposium on Theory of Computing |chapter=Quantum de finetti theorems under local measurements with applications |date=2013-01-01|series=STOC '13|location=New York, NY, USA|publisher=ACM|pages=861–870|arxiv=1210.6367|doi=10.1145/2488608.2488718|isbn=9781450320290|s2cid=1772280}}</ref> in topics like [[quantum key distribution]]<ref>{{cite arXiv|last=Renner|first=Renato|date=2005-12-30|title=Security of Quantum Key Distribution|arxiv=quant-ph/0512258}}</ref> and [[quantum entanglement|entanglement]] detection.<ref>{{Cite journal|last1=Doherty|first1=Andrew C.|last2=Parrilo|first2=Pablo A.|last3=Spedalieri|first3=Federico M.|date=2005-01-01|title=Detecting multipartite entanglement|journal=Physical Review A|volume=71|issue=3|pages=032333|arxiv=quant-ph/0407143|doi=10.1103/PhysRevA.71.032333|bibcode=2005PhRvA..71c2333D|s2cid=44241800}}</ref> A multivariate extension of de Finetti’s theorem can be used to derive [[Bose–Einstein statistics]] from the statistics of classical (i.e. independent) particles.<ref>{{cite journal |first1=A. |last1=Bach |first2=H. |last2=Blank |first3=H. |last3=Francke |title=Bose-Einstein statistics derived from the statistics of classical particles |journal=Lettere al Nuovo Cimento |volume=43 |pages=195–198 |year=1985 |issue=4 |doi=10.1007/BF02746978 |s2cid=121413539 }}</ref>