Difference between a mixture model and HHM

- If we examine a single time slice of the model, it can be seen as a mixture distribution with component densities given by
- It can be interpreted as an extension of a mixture model where the choice of mixture component for each observation is not independent but depends on the choice of component for the previous observations ()

Applications

- Speech recognition
- Natural language modeling
- On-line handwriting recognition
- analysis of biological sequences such as protein and DNA

Transition probability

- Latent variables; discrete multinomial variables = describe which component of the mixture is responsible for generating the corresponding observation
- The probability distribution of depends on the previous latent variable through conditional distribution
- Conditional distribution

- Inital latent node does not have a parent node, so it has a marginal distribution

- Lattice or trellis diagram

Emission probability

Example;

- Three Gaussian distribution/ two dice problem
- Handwriting