This tag is for questions relating to "Hidden Markov model", a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobservable (i.e. hidden) states.
The Hidden Markov model or, HMM is based on augmenting the Markov chain. It allows us to talk about both observed events (like words that we see in the input) and hidden events (like part-of-speech tags) that we think of as causal factors in our probabilistic model.
Definition: Let $~{\displaystyle X_{n}}~$ and $~{\displaystyle Y_{n}}~$ be discrete-time stochastic processes and $~{\displaystyle n\geq 1}~$. The pair $~{\displaystyle (X_{n},Y_{n})}~$ is a hidden markov model if
- $~{\displaystyle X_{n}}~$ is a Markov process and is not directly observable ("hidden");
- $$~{\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{1}=x_{1},\ldots ,X_{n}=x_{n}{\bigr )}=\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{n}=x_{n}{\bigr )},}$$ for every $~{\displaystyle n\geq 1,} {\displaystyle x_{1},\ldots ,x_{n},}~$ and an arbitrary measurable set $~{\displaystyle A}~$.
Applications: Hidden Markov models are especially known for their application in reinforcement learning and temporal pattern recognition such as speech, handwriting, gesture recognition, part-of-speech tagging, musical score following, partial discharges and bioinformatics.
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