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Equivalence and Reduction of Hidden Markov Models

By B. Vijay and Vijay Balasubramanian and Vijay Balasubramanian


Hidden Markov Models are one of the most popular and successful techniques used in statistical pattern recognition. However, they are not well understood on a fundamental level. For example, we do not knowhowtocharacterize the class of processes that can be well approximated by HMMs. This thesis tries to uncover the source of the intrinsic expressiveness of HMMs by studying when and whytwo models may represent the same stochastic process. Define two statistical models to be equivalent if they are models of exactly the same process. We use the theorems proved in this thesis to develop polynomial time algorithms to detect equivalence of prior distributions on an HMM, equivalence of HMMs and equivalence of HMMs with fixed priors. Wecharacterize Hidden Markov Models in terms of equivalence classes whose elements represent exactly the same processes and proceed to describe an algorithm to reduce HMMs to essentially unique and minimal, canonical representations. These canonical forms are essentially "smallest representatives" of their equivalence classes, and the number of parameters describing them can be considered a representation for the complexityofthestochastic process they model. On the waytodeveloping our reduction algorithm, we define Generalized Markov Models which relax the positivity constraint on HMM parameters. This generalization is derived by taking the view that an interpretation of model parameters as probabilities is less important than a parsimonious representation of stochastic processes

Topics: Title, Professor, Arti cial Intelligence La
Year: 1993
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