Hidden Markov Model

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

The Transition Model

P(Xt∣Xt−1)P(X_t\mid X_{t-1})

is a single S×SS\times S matrix.

The Sensor Model P(et∣Xt=i)P(e_t\mid X_t=i) is an SS-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

f1:t+1=α⋅Ot+1Ttf1:t\mathrm{f}_{1: t+1}=\alpha \cdot \mathrm{O}_{t+1} T^t \mathrm{f}_{1: t}

Smoothing

bk+1:t=TOk+1 bk+2:t\mathrm{b}_{k+1: t}=\mathrm{TO}_{k+1} \mathrm{~b}_{k+2: t}

Zustandsraum Emissionen

Sensor Model Anfangsverteilung

Beispiel Veranschaulichung

Bildschirmfoto 2022-03-09 um 20.42.17.png