Editing Hidden Markov model (section)

== 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, a probability measure can be imposed 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>. By ignoring the distinction between <math>B_1, B_2</math>, this space of subshifts is projected 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 \mid 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 |last1=Boyle |first1=Mike |title=Hidden Markov processes in the context of symbolic dynamics |date=2010-01-13 |last2=Petersen |first2=Karl|arxiv=0907.1858}}</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"/>).