Product Estimators for Hidden Markov Models

Abstract

In this thesis, risk-sensitive estimation for Hidden Markov Models isstudied from a dynamical systems point of view. We show that risk-sensitive estimators belong to a broaderclass of product estimators in which risk-sensitivity willbe shown to be related to certain scaling functions.The product structureand the scaling functions perspective give us new insights into the underlying mechanism of 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 exposerelations among the transition probabilities, risk-sensitivity andthe decision regions. We introduce the risk-sensitive Maximum A Posterior Probability (MAP) criterion for HMM's with discrete rangeobservation. This criterion is the discrete time finite dimensionalversion 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-sensitivityon the behavior of the sample paths of the estimator; theproduct structure representationwill play an important role.