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In statistics, an expectation–maximization (EM) algorithm is an iterative method for finding maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the ''E'' step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step. ==History== The EM algorithm was explained and given its name in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin.〔 〕 They pointed out that the method had been "proposed many times in special circumstances" by earlier authors. In particular, a very detailed treatment of the EM method for exponential families was published by Rolf Sundberg in his thesis and several papers〔 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.〕 following his collaboration with Per Martin-Löf and Anders Martin-Löf.〔See the acknowledgement by Dempster, Laird and Rubin on pages 3, 5 and 11.〕〔G. Kulldorff. 1961.'' Contributions to the theory of estimation from grouped and partially grouped samples''. Almqvist & Wiksell.〕〔Anders Martin-Löf. 1963. "Utvärdering av livslängder i subnanosekundsområdet" ("Evaluation of sub-nanosecond lifetimes"). ("Sundberg formula")〕〔 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).〕〔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")〕〔 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, A. P. Dempster, D. Basu, D. R. Cox, A. W. F. Edwards, D. A. Sprott, G. A. Barnard, O. Barndorff-Nielsen, J. D. Kalbfleisch and 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.〕〔 Martin-Löf, Per The notion of redundancy and its use as a quantitative measure of the discrepancy between a statistical hypothesis and a set of observational data. ''Scand. J. Statist.'' 1 (1974), no. 1, 3–18.〕 The Dempster-Laird-Rubin paper in 1977 generalized the method and sketched a convergence analysis for a wider class of problems. Regardless of earlier inventions, the innovative Dempster-Laird-Rubin paper in the ''Journal of the Royal Statistical Society'' received an enthusiastic discussion at the Royal Statistical Society meeting with Sundberg calling the paper "brilliant". The Dempster-Laird-Rubin paper established the EM method as an important tool of statistical analysis. The convergence analysis of the Dempster-Laird-Rubin paper was flawed and a correct convergence analysis was published by C.F. Jeff Wu in 1983.〔 〕 Wu's proof established the EM method's convergence outside of the exponential family, as claimed by Dempster-Laird-Rubin.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Expectation–maximization algorithm」の詳細全文を読む スポンサード リンク
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