Learning in Hidden Markov Models

Author(s)

Roberto Castro Alamino

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2005-11-10

Examining board members

Nestor Felipe Caticha Alfonso; Marco Aurelio Pires Idiart; Carmen Pimentel Cintra do Prado; Rogério Rosenfeld; Renato Vicente

• Advisor: Nestor Felipe Caticha Alfonso

Abstract

In this work we study learning in a specific class of probabilistic models known as Hidden Markov Models (HMMs). First we discuss its basic theory and after we make a detailed study of the behavior of five different learning algorithms, two of them already known in the literature and the other three proposed by us in this work. The five algorithms are described below in the sequence they are presented in the thesis: Baum-Welch Algorithm(BW): consists of a renowed offline algorithm obtained by applying the EM-algorithm to the particular case of HMMs. Through the literature it is common to refer to it by the name Baum-Welch Reestimation Formulas. Baum-Welch Online Algorithm (BWO): online version of BW proposed by us. Baldi-Chauvin Algorithm (BC): online algorithm proposed by Baldi and Chauvin in [5] where a softmax representation for the probabilities of the HMMs is used and where the aim is to maximize the model likelihood at each iteration step. Online Bayesian Algorithm (BKL): an algorithm developed by us based on the work of Opper [74] where, after updating the probability distribution of the model with each new data, the obtained density is projected into a parametric family of tractable distributions minimizing the Kullback-Leibler distance between both. Mean Posterior Algorithm (PM): a simplification of BKL where the projection after the update is made on the mean posterior distribution. For each one of the above algorithms, we obtain learning curves by means of simulations where we use two distinct measures of generalization error: the Kullback-Leibler distance (dKL) and the Euclidian distance (dE). With exception of the BW algorithm, which can be used only in offline learning situations, we study for all the other algorithms the learning curves for both learning situations: online and offiine. We compare the performance of the algorithms with one another and discuss the results showing that, besides its larger computation time, the bayesian algorithm PM, proposed by us, is superior to the other non-bayesian algorithms with respect to the generalization in static learning situations and that it has a performance that is very close to the bayesian algorithm BKL. We also make a comparison between algorithms PM and BC in learning situations that change with time using artificially generated data and in one situation with real data, with a simplified scenario, where we use them to predict the behavior of the São Paulo Stock Market Index (BOVESPA) showing that, although they need a large learning period, after that initial phase the predictions obtained by both algorithms are surprisingly good. Finally, we present a discussion about learning and symmetry breaking based on the presented studies. (AU)