Lumpable hidden Markov models - model reduction and reduced complexity filtering

Copyright © 2000 IEEEThis paper is concerned with filtering of hidden Markov processes (HMP) which possess (or approximately possess) the property of lumpability. This property is a generalization of the property of lumpability of a Markov chain which has been previously addressed by others. In essence, the property of lumpability means that there is a partition of the (atomic) states of the Markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned. We prove necessary and sufficient conditions on the HMP for exact lumpability to hold. For a particular class of hidden Markov models (HMM), namely finite output alphabet models, conditions for lumpability of all HMP representable by a specified HMM are given. The corresponding optimal filter algorithms for the aggregated states are then derived. The paper also describes an approach to efficient suboptimal filtering for HMP which are approximately lumpable. By this we mean that the HMM generating the process may be approximated by a lumpable HMM. This approach involves directly finding a lumped HMM which approximates the original HMM well, in a matrix norm sense. An alternative approach for model reduction based on approximating a given HMM by an exactly lumpable HMM is also derived. This method is based on the alternating convex projections algorithm. Some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithmsLangford B. White, Robert Mahony and Gary D. Brush

Localizing the Latent Structure Canonical Uncertainty: Entropy Profiles for Hidden Markov Models

This report addresses state inference for hidden Markov models. These models rely on unobserved states, which often have a meaningful interpretation. This makes it necessary to develop diagnostic tools for quantification of state uncertainty. The entropy of the state sequence that explains an observed sequence for a given hidden Markov chain model can be considered as the canonical measure of state sequence uncertainty. This canonical measure of state sequence uncertainty is not reflected by the classic multivariate state profiles computed by the smoothing algorithm, which summarizes the possible state sequences. Here, we introduce a new type of profiles which have the following properties: (i) these profiles of conditional entropies are a decomposition of the canonical measure of state sequence uncertainty along the sequence and makes it possible to localize this uncertainty, (ii) these profiles are univariate and thus remain easily interpretable on tree structures. We show how to extend the smoothing algorithms for hidden Markov chain and tree models to compute these entropy profiles efficiently.Comment: Submitted to Journal of Machine Learning Research; No RR-7896 (2012