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→History
==History==
==History==
==History==
The EM algorithm was explained and given its name in a classic 1977 paper by [[Arthur P. Dempster|Arthur Dempster]], [[Nan Laird]], and [[Donald Rubin]].<ref name="Dempster1977">
{{cite journal
|last1=Dempster |first1= A.P. |author-link1=Arthur P. Dempster
|last2=Laird |first2=N.M. |author-link2=Nan Laird |last3=Rubin
|first3=D.B. |author-link3=Donald Rubin
|title=Maximum Likelihood from Incomplete Data via the EM Algorithm
|journal=[[Journal of the Royal Statistical Society, Series B]]
|year=1977 |volume=39 |issue=1 |pages=1–38
|jstor=2984875 |mr= 0501537
}}
</ref> They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. One of the earliest is the gene-counting method for estimating allele frequencies by [[Cedric Smith (statistician)|Cedric Smith]].<ref>{{cite journal |last1=Ceppelini |first1=R.M. |title=The estimation of gene frequencies in a random-mating population |journal=Ann. Hum. Genet. |volume=20 |issue=2 |pages=97–115 |doi=10.1111/j.1469-1809.1955.tb01360.x|pmid=13268982 |year=1955 |s2cid=38625779 }}</ref> A very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers<ref name="Sundberg1974">{{cite journal
|last=Sundberg |first=Rolf
|title=Maximum likelihood theory for incomplete data from an exponential family
|journal=Scandinavian Journal of Statistics
|volume=1 |year=1974 |issue=2 |pages=49–58
|jstor=4615553 |mr= 381110
}}</ref><ref name="Sundberg1971">
Rolf Sundberg. 1971. ''Maximum likelihood theory and applications for distributions generated when observing a function of an exponential family variable''. Dissertation, Institute for Mathematical Statistics, Stockholm University.</ref><ref name="Sundberg1976">{{cite journal
|last=Sundberg |first=Rolf
|year=1976
|title=An iterative method for solution of the likelihood equations for incomplete data from exponential families
|journal=Communications in Statistics – Simulation and Computation
|volume=5 |issue=1 |pages=55–64
|doi=10.1080/03610917608812007 |mr=443190
}}</ref> following his collaboration with [[Per Martin-Löf]] and [[Anders Martin-Löf]].<ref>See the acknowledgement by Dempster, Laird and Rubin on pages 3, 5 and 11.</ref><ref>G. Kulldorff. 1961.'' Contributions to the theory of estimation from grouped and partially grouped samples''. Almqvist & Wiksell.</ref><ref name="Martin-Löf1963">Anders Martin-Löf. 1963. "Utvärdering av livslängder i subnanosekundsområdet" ("Evaluation of sub-nanosecond lifetimes"). ("Sundberg formula")</ref><ref name="Martin-Löf1966">[[Per Martin-Löf]]. 1966. ''Statistics from the point of view of statistical mechanics''. Lecture notes, Mathematical Institute, Aarhus University. ("Sundberg formula" credited to Anders Martin-Löf).</ref><ref name="Martin-Löf1970">[[Per Martin-Löf]]. 1970. ''Statistika Modeller (Statistical Models): Anteckningar från seminarier läsåret 1969–1970 (Notes from seminars in the academic year 1969-1970), with the assistance of Rolf Sundberg.'' Stockholm University. ("Sundberg formula")</ref><!-- * Martin-Löf, P. "Exact tests, confidence regions and estimates", with a discussion by [[A. W. F. Edwards]], [[George A. Barnard|G. A. Barnard]], D. A. Sprott, O. Barndorff-Nielsen, [[D. Basu]] and [[Rasch model|G. Rasch]]. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 121–138. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974. --><ref name="Martin-Löf1974a">Martin-Löf, P. The notion of redundancy and its use as a quantitative measure of the deviation between a statistical hypothesis and a set of observational data. With a discussion by F. Abildgård, [[Arthur P. Dempster|A. P. Dempster]], [[D. Basu]], [[D. R. Cox]], [[A. W. F. Edwards]], D. A. Sprott, [[George A. Barnard|G. A. Barnard]], O. Barndorff-Nielsen, J. D. Kalbfleisch and [[Rasch model|G. Rasch]] and a reply by the author. ''Proceedings of Conference on Foundational Questions in Statistical Inference'' (Aarhus, 1973), pp. 1–42. Memoirs, No. 1, Dept. Theoret. Statist., Inst. Math., Univ. Aarhus, Aarhus, 1974.</ref><ref name="Martin-Löf1974b">{{cite journal |last=Martin-Löf |first=Per |title=The notion of redundancy and its use as a quantitative measure of the discrepancy between a statistical hypothesis and a set of observational data |journal=Scand. J. Statist. |volume=1 |year=1974 |issue=1 |pages=3–18 }}</ref> The Dempster–Laird–Rubin paper in 1977 generalized the method and sketched a convergence analysis for a wider class of problems. The Dempster–Laird–Rubin paper established the EM method as an important tool of statistical analysis.
The convergence analysis of the Dempster–Laird–Rubin algorithm was flawed and a correct convergence analysis was published by [[C. F. Jeff Wu]] in 1983.<ref name="Wu">
{{cite journal
|first=C. F. Jeff
|last=Wu
|title=On the Convergence Properties of the EM Algorithm
|journal=[[Annals of Statistics]]
|volume=11
|issue=1
|date=Mar 1983
|pages=95–103
|jstor=2240463
|doi=10.1214/aos/1176346060
|mr= 684867
|doi-access=free
}}</ref>
Wu's proof established the EM method's convergence outside of the [[exponential family]], as claimed by Dempster–Laird–Rubin.<ref name="Wu" />
The EM algorithm was explained and given its name in a classic 1977 paper by [[Arthur P. Dempster|Arthur Dempster]], [[Nan Laird]], and [[Donald Rubin]].<ref name="Dempster1977">
The EM algorithm was explained and given its name in a classic 1977 paper by [[Arthur P. Dempster|Arthur Dempster]], [[Nan Laird]], and [[Donald Rubin]].<ref name="Dempster1977">