Applications of Hidden Markov Chains

Let $X_{n}$ and $Y_{n}$ be discrete-time stochastic processes and $n\geq 1$. The pair $(X_{n},Y_{n})$ is a hidden Markov model if

• $X_{n}$ is a Markov process whose behavior is not directly observable ("hidden");
• $\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{1}=x_{1},\ldots ,X_{n}=x_{n}{\bigr )}=\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{n}=x_{n}{\bigr )}$,

Let $X_{t}$ and $Y_{t}$ be continuous-time stochastic processes. The pair $(X_{t},Y_{t})$ is a hidden Markov model if

• $X_{t}$ is a Markov process whose behavior is not directly observable ("hidden");
• $\operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid \{X_{t}\in B_{t}\}_{t\leq t_{0}})=\operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid X_{t_{0}}\in B_{t_{0}})$,

Terminology

The states of the process $X_{n}$(resp. $X_{t})$are called hidden states, and $\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\mid X_{n}=x_{n}{\bigr )}$(resp. $\operatorname {\mathbf {P} } {\bigl (}Y_{t}\in A\mid X_{t}\in B_{t}{\bigr )})$ is called emission probability or output probability.