Editing Hidden Markov model (section)

== History ==
Hidden Markov models were described in a series of statistical papers by [[Leonard E. Baum]] and other authors in the second half of the 1960s.<ref>{{cite journal |last=Baum |first=L. E. |author2=Petrie, T. |title=Statistical Inference for Probabilistic Functions of Finite State Markov Chains |journal=The Annals of Mathematical Statistics |year=1966 |volume=37 |issue=6 |pages=1554–1563 |doi=10.1214/aoms/1177699147|doi-access=free}}</ref><ref>{{Cite journal |last1=Baum |first1=L. E. |last2=Eagon |first2=J. A. |doi=10.1090/S0002-9904-1967-11751-8 |title=An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology |journal=[[Bulletin of the American Mathematical Society]] |volume=73 |issue=3 |pages=360 |year=1967 |zbl=0157.11101 |url=http://projecteuclid.org/euclid.bams/1183528841 |doi-access=free}}</ref><ref>{{cite journal |last=Baum |first=L. E. |author2=Sell, G. R. |title=Growth transformations for functions on manifolds |journal=Pacific Journal of Mathematics |year=1968 |volume=27 |issue=2 |pages=211–227 |doi=10.2140/pjm.1968.27.211 |doi-access=free}}</ref><ref>{{Cite journal |last1=Baum |first1=L. E. |author-link1=Leonard E. Baum |last2=Petrie |first2=T. |last3=Soules |first3=G. |last4=Weiss |first4=N. |title=A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains |doi=10.1214/aoms/1177697196 |journal=[[The Annals of Mathematical Statistics]] |volume=41 |issue=1 |pages=164–171 |year=1970 |jstor=2239727 |zbl=0188.49603 |mr=287613 |doi-access=free}}</ref><ref>{{cite journal |last=Baum |first=L.E. |title=An Inequality and Associated Maximization Technique in Statistical Estimation of Probabilistic Functions of a Markov Process |journal=Inequalities |year=1972 |volume=3 |pages=1–8}}</ref> One of the first applications of HMMs was [[speech recognition]], starting in the mid-1970s.<ref>{{Cite journal |last1=Baker |first1=J. |author-link1=James K. Baker |doi=10.1109/TASSP.1975.1162650 |title=The DRAGON system—An overview |journal=IEEE Transactions on Acoustics, Speech, and Signal Processing |volume=23 |pages=24–29 |year=1975}}</ref><ref>{{Cite journal |last1=Jelinek |first1=F. |last2=Bahl |first2=L. |last3=Mercer |first3=R. |doi=10.1109/TIT.1975.1055384 |title=Design of a linguistic statistical decoder for the recognition of continuous speech |journal=[[IEEE Transactions on Information Theory]] |volume=21 |issue=3 |pages=250 |year=1975}}</ref><ref>{{cite book |title=Hidden Markov Models for Speech Recognition |publisher=Edinburgh University Press |year=1990 |isbn=978-0-7486-0162-2 |author1=Xuedong Huang |author2=M. Jack |author3=Y. Ariki |author-link1=Xuedong Huang}}</ref><ref>{{cite book |title=Spoken Language Processing |publisher=Prentice Hall |year=2001 |isbn=978-0-13-022616-7 |author1=Xuedong Huang |author2=Alex Acero |author3=Hsiao-Wuen Hon |author-link1=Xuedong Huang}}</ref> From the linguistics point of view, hidden Markov models are equivalent to stochastic regular grammar.<ref>{{Cite book |last1=Carrasco |first1=Rafael C. |last2=Oncina |first2=Jose |date=1994 |editor-last=Carrasco |editor-first=Rafael C. |editor2-last=Oncina |editor2-first=Jose |chapter=Learning stochastic regular grammars by means of a state merging method |chapter-url=https://link.springer.com/chapter/10.1007/3-540-58473-0_144 |title=Grammatical Inference and Applications |series=Lecture Notes in Computer Science |volume=862 |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=139–152 |doi=10.1007/3-540-58473-0_144 |isbn=978-3-540-48985-6}}</ref>

In the second half of the 1980s, HMMs began to be applied to the analysis of biological sequences,<ref>{{cite journal |doi=10.1016/0022-2836(86)90289-5 |author=M. Bishop and E. Thompson |title=Maximum Likelihood Alignment of DNA Sequences |journal=[[Journal of Molecular Biology]] |volume=190 |issue=2 |pages=159–165 |year=1986 |pmid=3641921}} {{subscription required}} {{closed access}}</ref> in particular [[DNA]]. Since then, they have become ubiquitous in the field of [[bioinformatics]].<ref name=durbin>{{Durbin 1998|mode=cs1}}</ref>