Gábor Molnár-Sáska

New methods for the statistical analysis of Hidden Markov Models

The estimation of Hidden Markov Models has attracted a lot of attention recently. The purpose of this paper is to lay the foundation for a new approach for the analysis of the maximum-likelihood estimation of HMM-s, using representation of HMM-s due to Borkar (1993). A useful connection between the estimation theory of HMM-s and linear stochastic systems is established via the theory of L-mixing processes. The results are potentially useful for deriving strong approximation results, which are in turn applicable to analyze adaptive predictors and change detection methods.

A new method for the analysis of Hidden Markov model estimates

Abstract The estimation of Hidden Markov Models has attracted a lot of attention recently, see results of (Le Gland and Mevel, 2000), (Leroux, 1992), (Mevel and Finesso, 2000). The purpose of this paper is to lay the foundation for a new approach for the analysis of the maximum-likelihood estimation of HMM-s, using representation of HMM-s due to (Borkar, 1993). Useful connection between the estimation theory of HMM-s and linear stochastic systems is established via the theory of L-mixing processes developed in (Gerencser, 1988).

Recursive estimation of Hidden Markov Models

A recursive estimation method for Hidden Markov Models has been proposed in [24]. As suggested there the proposed recursive algorithm could be analyzed via the theory of stochastic approximations developed in [4]. The purpose of this note is to verify the basic probabilistic conditions of [4], given in Part II, Chapter 1 of [4]. For this purpose we consider a general class of Markov models in which a simple Markov process is passed through an exponentially stable non-linear system. The general theory is relatively easily applied to HMMs extended by their filter process and their derivatives, see [1].

Change detection of hidden Markov models

Let /spl theta//spl circ//sub N/ denote the maximum-likelihood estimator of the true parameter /spl theta/* of a hidden Markov Model with fixed gain or forgetting rate /spl lambda/. We establish an explicit formula for the error term /spl theta//spl circ//sub N/ - /spl theta/*. Using the representation of the error term we investigate the effect of parameter uncertainty on the performance of an adaptive encoding procedure. Using this result and ideas from the theory of stochastic complexity a change point detection method for HMM-s will be developed.

Identification of hidden markov models - : Uniform LLN-s

We consider hidden Markov processes in discrete time with a finite state space X and a general observation or read-out space Y. The identification of the unknown dynamics is carried out by the conditional maximum-likelihood method. The normalized log-likelihood function is shown to satisfy a uniform law of large numbers over certain compact subsets of the parameter space. Two cases are covered: first, when the running value of the transition probability matrix, denoted by Q is positive, second, when Q is primitive, but the read-out densities are strictly positive.