Applications of Hidden Markov Chains

Definition

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");
$\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{1}=x_{1},\ldots ,X_{n}=x_{n}{\bigr )}=\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{n}=x_{n}{\bigr )}$ ,

for every $n\geq 1$ , $x_{1},\ldots ,x_{n}$ , and every Borel set $A$ .

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");
$\operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid \{X_{t}\in B_{t}\}_{t\leq t_{0}})=\operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid X_{t_{0}}\in B_{t_{0}})$ ,

for every $t_{0}$ , every Borel set $A$ , and every family of Borel sets $\{B_{t}\}_{t\leq t_{0}}$ .

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.