Local Equivalence Problem in Hidden Markov Model
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In the hidden Markovian process, there is a possibility that two different transition matrices for hidden and observed variables yield the same stochastic behavior for the observed variables. Since such two transition matrices cannot be distinguished, we need to identify them and consider that they are equivalent, in practice. We address the equivalence problem of hidden Markovian process in a local neighborhood by using the geometrical structure of hidden Markovian process. For this aim, we formulate an exponential family of Y-valued transition matrices by using generators. Then, the above equivalence problem is formulated as the equivalence problem of generators. Taking account into this equivalence problem, we derive several concrete parametrizations in several natural cases.
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