Infinite Hidden Markov Model

Motivation

One commonly encountered model in Bayesian inference is the Hidden Markov Model (HMM). A potential problem with HMMs is that they require specifying in advance a finite number of (discrete) latent states, whereas we may not know the total number of latent states. Instead, we’d prefer to use a non-parametric model that grows in complexity with the amount of observed data.

To meet this need, Beal, Ghahramani and Rasumussen 2001 introduced the infinite Hidden Markov Model (iHMM), also known as the Hierarchical Dirichlet Process-Hidden Markov Model (HDP-HMM). This model modifies the traditional HMM by allowing for new latent states and new emission characters to be added as more data is observed. To make inference tractable, the authors integrate out the infinite possible parameters to obtain a posterior over the latent states and posteriors over three hyperparameters:

1. \(\alpha\), which controls the tendency to linger in a state

2. \(\beta\), which controls the tendency to explore new transitions/states

3. \(\gamma\), which controls the expected number of hidden states

Technical Note: I think Hierarchical Dirichlet Process is a misnomer. Due to that particular integration of parameters, the iHMM is more similar to the Chinese Restaurant Process than the Dirichlet Process.

Definition

Inference