Large Deviation Bounds for Functionals of Viterbi Paths

In a number of applications, the underlying stochastic process is modeled as a finite-state discrete-time Markov chain that cannot be observed directly and is represented by an auxiliary process. The maximum a posteriori (MAP) estimator is widely used to estimate states of this hidden Markov model through available observations. The MAP path estimator based on a finite number of observations is calculated by the Viterbi algorithm, and is often referred to as the Viterbi path. It was recently shown in, and, (see also and) that under mild conditions, the sequence of estimators of a given state converges almost surely to a limiting regenerative process as the number of observations approaches infinity. This in particular implies a law of large numbers for some functionals of hidden states and finite Viterbi paths. The aim of this paper is to provide the corresponding large deviation estimates.

[1]  J. Lember,et al.  ADJUSTED VITERBI TRAINING , 2004, Probability in the Engineering and Informational Sciences.

[2]  Lawrence R. Rabiner,et al.  A tutorial on hidden Markov models and selected applications in speech recognition , 1989, Proc. IEEE.

[3]  Neri Merhav,et al.  Hidden Markov processes , 2002, IEEE Trans. Inf. Theory.

[4]  E. Çinlar.  Markov additive processes. II , 1972 .

[5]  Eric Moulines,et al.  Inference in hidden Markov models , 2010, Springer series in statistics.

[6]  Amke Caliebe,et al.  Properties of the maximum a posteriori path estimator in hidden Markov models , 2006, IEEE Transactions on Information Theory.

[7]  T. Kuczek,et al.  A large-deviation result for regenerative processes , 1991 .

[8]  J. Lember,et al.  Adjusted Viterbi training for hidden Markov models , 2007, 0709.2317.

[9]  Simon Kasif,et al.  Computational methods in molecular biology , 1998 .

[10]  Haikady N. Nagaraja,et al.  Inference in Hidden Markov Models , 2006, Technometrics.

[11]  Frederick Jelinek,et al.  Statistical methods for speech recognition , 1997 .

[12]  Biing-Hwang Juang,et al.  Hidden Markov Models for Speech Recognition , 1991 .

[13]  Uwe Rösler,et al.  Convergence of the maximum a posteriori path estimator in hidden Markov models , 2002, IEEE Trans. Inf. Theory.

[14]  James A. Bucklew.  The Blind Simulation Problem and Regenerative Processes , 1998, IEEE Trans. Inf. Theory.

[15]  Sean R. Eddy,et al.  Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids , 1998 .

[16]  Vladimir Kalashnikov,et al.  Topics on Regenerative Processes , 1994 .

[17]  Jüri Lember,et al.  A Constructive Proof of the Existence of Viterbi Processes , 2008, IEEE Transactions on Information Theory.

[18]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[19]  Jr. G. Forney,et al.  Viterbi Algorithm , 1973, Encyclopedia of Machine Learning.

[20]  P. Ney,et al.  MARKOV ADDITIVE PROCESSES II. LARGE DEVIATIONS , 1987 .