Hidden Markov Model

A hidden Markov model is a Markov Chain with a single discrete state variable $X_t$ with domain $\{1,...,S\}$ and a single, discrete evidence variable.

The Transition Model $P(X_t\mid X_{t-1})$ is a single $S\times S$ matrix.

The Sensor Model $P(e_t\mid X_t=i)$ is an $S$-vector.

The idea is that we can now formulate Markov Inference as matrix calculations (which can be sped up by optimized algorithms).

To make this work we create a diagonal sensor matrix from the Sensor Model vector.

Filtering ${\mathrm{f}}_{1:t+1}=\alpha \cdot {\mathrm{O}}_{t+1}{T}^t{\mathrm{f}}_{1:t}$

Smoothing ${\mathrm{b}}_{k+1:t}={\mathrm{TO}}_{k+1}{\mathrm{b}}_{k+2:t}$

Zustandsraum Emissionen

Sensor Model Anfangsverteilung

Beispiel Veranschaulichung