# Hierarchical hidden Markov model

Hierarchical hidden Markov model

The Hierarchical hidden Markov model (HHMM) is a statistical model derived from the hidden Markov model (HMM). In an HHMM each state is considered to be a self contained probabilistic model. More precisely each stateof the HHMM is itself an HHMM.

Background

It is sometimes useful to use HMMs in specific structures in order to facilitate learning and generalization. For example, even though a fully connected HMM could always be used if enough training data isavailable it is often useful to constrain the model by not allowingarbitrary state transitions. In the same way it can be beneficial toembed the HMM into a greater structure; which, theoretically, may notbe able to solve any other problems than the basic HMM but can solvesome problems more efficiently when it comes to the amount of trainingdata required.

The Hierarchical Hidden Markov Model

In the hierarchical hidden Markov model (HHMM) each state is consideredto be a self contained probabilistic model. More precisely each stateof the HHMM is itself an HHMM. This implies that the states of the HHMMemits sequences of observation symbols rather than singleobservation symbols as is the case for the standard HMM states.

When a state in an HHMM is activated, it will activate its own probabilisticmodel, i.e. it will activate one of the states of the underlying HHMM,which in turn may activate its underlying HHMM and so on. The processis repeated until a special state, called a production state, isactivated. Only the production states emit observation symbols inthe usual HMM sense. When the production state has emitted a symbol,control returns to the state that activated the production state.The states that do not directly emit observations symbols are calledinternal states. The activation of a state in an HHMM under an internal state is called a "vertical transition". After a vertical transition is completed a "horizontal transition"occurs to a state within the same level. When a horizontaltransition leads to a "terminating" state control is returned tothe state in the HHMM, higher up in the hierarchy, that produced thelast vertical transition.

Remember that a vertical transition can result in more verticaltransitions before reaching a sequence of production states andfinally returning to the top level. Thus the production states visitedgives rise to a sequence of observation symbols that is "produced"by the state at the top level.

The methods for estimating the HHMM parameters and model structure are more complex than for the HMM and the interested reader is referred to(Fine "et al.", 1998).

It should be pointed out that the HMM and HHMM belong to thesame class of classifiers. That is, they can be used to solve thesame set of problems. In fact, the HHMM can betransformed into a standard HMM. However, the HHMM utilizes its structure to solve a subset of the problems moreefficiently.

* Layered hidden Markov model
* Hierarchical Temporal Memory

References

S. Fine, Y. Singer and N. Tishby, "The Hierarchical Hidden Markov Model: Analysis and Applications", Machine Learning, vol. 32, p. 41-62, 1998

K.Murphy and M.Paskin. "Linear Time Inference in Hierarchical HMMs", NIPS-01 (Neural Info. Proc. Systems).

H. Bui, D. Phung and S. Venkatesh. "Hierarchical Hidden Markov Models with General State Hierarchy", AAAI-04 (National Conference on Artificial Intelligence).

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