Contents

Inference in hidden markov models¶

The main idea is how to perform marginal inference on the hidden states given observed. Or perform parameter estimation and so on.

Forward-Backward¶

For a hidden markov model of a form:

We are given:

  • initial probabilities \(p(z_1)\)

  • transition probabilities \(p(z_{k+1}|z_k)\)

  • emission probabilities \(p(x_k|z_k)\)

The main goal of the Forward-Backward algorithm is to compute the marginals of form:

\[ p(z_k|x_{1:n}) \propto p(z_k,x_{1:n}) = p(x_{k+1:n}|z_k)p(z_k,x_{1:k}) \]

This estimation consists of two steps:

  1. Forward pass \(p(z_k,x_{1:k})\)

  2. Backward pass \(p(x_{k+1:n}|z_k)\)

Example:

Forward¶

Backward¶

MAP estimation (Viberty)¶

Kernel machines SVM classification