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The model suitable in the context of longitudinal data is named latent Markov model.<ref>{{Cite book|title=Panel Analysis: Latent Probability Models for Attitude and Behaviour Processes|last=Wiggins|first=L. M.|publisher=Elsevier|year=1973|___location=Amsterdam}}</ref> The basic version of this model has been extended to include individual covariates, random effects and to model more complex data structures such as multilevel data. A complete overview of the latent Markov models, with special attention to the model assumptions and to their practical use is provided in<ref>{{Cite book|url=https://sites.google.com/site/latentmarkovbook/home|title=Latent Markov models for longitudinal data|last1=Bartolucci|first1=F.|last2=Farcomeni|first2=A.|last3=Pennoni|first3=F.|publisher=Chapman and Hall/CRC|year=2013|isbn=978-14-3981-708-7|___location=Boca Raton}}</ref>
== Measure theory ==
{{See also|Subshift of finite type}}
[[File:Blackwell_HMM_example.png|thumb|193x193px|The hidden part of a hidden Markov model, whose observable states is non-Markovian.]]
Given a Markov transition matrix and an invariant distribution on the states, we can impose a probability measure on the set of subshifts. For example, consider the Markov chain given on the left on the states <math>A, B_1, B_2 </math>, with invariant distribution <math>\pi = (2/7, 4/7, 1/7) </math>. If we "forget" the distinction between <math>B_1, B_2</math>, we project this space of subshifts on <math>A, B_1, B_2 </math> into another space of subshifts on <math>A, B </math>, and this projection also projects the probability measure down to a probability measure on the subshifts on <math>A, B </math>.
The curious thing is that the probability measure on the subshifts on <math>A, B </math> is not created by a Markov chain on <math>A, B </math>, not even multiple orders. Intuitively, this is because if one observes a long sequence of <math>B^n</math>, then one would become increasingly sure that the <math>Pr(A | B^n) \to \frac 23 </math>, meaning that the observable part of the system can be affected by something infinitely in the past.<ref name=":0">''[https://web.archive.org/web/20221005013617/https://petersen.web.unc.edu/wp-content/uploads/sites/17054/2018/04/Main.pdf Sofic Measures: Characterizations of Hidden Markov Chains by Linear Algebra, Formal Languages, and Symbolic Dynamics]'' - Karl Petersen, Mathematics 210, Spring 2006, University of North Carolina at Chapel Hill</ref><ref name=":1">{{Citation |last=Boyle |first=Mike |title=Hidden Markov processes in the context of symbolic dynamics |date=2010-01-13 |url=http://arxiv.org/abs/0907.1858 |access-date=2024-02-07 |doi=10.48550/arXiv.0907.1858 |last2=Petersen |first2=Karl}}</ref>
Conversely, there exists a space of subshifts on 6 symbols, projected to subshifts on 2 symbols, such that any Markov measure on the smaller subshift has a preimage measure that is not Markov of any order (Example 2.6 <ref name=":1" />).
== See also ==
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