Product estimators for hidden Markov models

In this thesis, risk-sensitive estimation for the Hidden Markov Models is studied from a dynamical systems point of view. We show that risk-sensitive estimators belong to a broader class of product estimators in which risk-sensitivity will be shown to be related to certain scaling functions. The product structure and the scaling functions perspective give us new insights into the underlying mechanism of the risk-sensitive estimation. For the first time, in a series of theorems and examples, we relate risk-sensitivity to the dynamics of the underlying process and expose relations among the transition probabilities, risk-sensitivity and the decision regions. We introduce the risk-sensitive Maximum A Posterior Probability (MAP) criterion for HMM's with discrete range observation. This criterion is the discrete time finite dimensional version of the classic risk-sensitive estimation problem for linear/quadratic partial observation case.; The risk-sensitive filters take into account the “higher order” moments of the of the estimation error. In the context of risk-sensitive MAP for HMM's, we clarify and quantify the influence of risk-sensitivity on the behavior of the sample paths of the estimator; the product structure representation will play an important role.