Hidden Markov Models

Table of contents

  1. Definition
  2. Filtering
  3. Smoothing
  4. Viterbi

Definition

  • A set of discrete or continuous variables \(v_1, \dots, v_T = v_{1:T}\) which represent the observed time-series
  • A set of discrete hidden variables \(h_{1:T}\) that generate the observations
\[p(v_{1:T}, h_{1:T}) = p(v_1 \mid h_1)p(h_1) \prod_{t=2}^T p(v_t \mid h_t)p(h_t \mid h_{t-1})\]

The past is independent of the future given the present!

HMM

Filtering

Infer \(h_t\) from \(p(h_t \mid v_{1:t})\) which uses the observations up to time \(t\) This is carried out by passing messages up and to the right.

Smoothing

Infer \(h_t\) from \(p(h_t \mid v_{1:T})\) which also uses future observations Combine filtering messages with messages up and to the left.

Viterbi

Infer the most likely hidden sequence \(h_{1:T}\) from \(\arg \max_{h_{1:T}} p(h_{1:T} \mid v_{1:T})\)