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.