Exponential Bounds for Convergence of Entropy Rate Approximations in Hidden Markov Models Satisfying a Path-Mergeability Condition

A hidden Markov model (HMM) is said to have path-mergeable states if for any two states i,j there exists a word w and state k such that it is possible to transition from both i and j to k while emitting w. We show that for a finite HMM with path-mergeable states the block estimates of the entropy rate converge exponentially fast. We also show that the path-mergeability property is asymptotically typical in the space of HMM topolgies and easily testable.

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

[2]  DISCRETE-TIME FILTERING FOR LINEAR SYSTEMS IN CORRELATED NOISE WITH NON-GAUSSIAN INITIAL CONDITIONS: FORMULAS AND ASYMPTOTICS , 1989 .

[3]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[4]  John J. Birch.  Approximations for the Entropy for Functions of Markov Chains , 1962 .

[5]  R. Liptser,et al.  Stability of nonlinear filters in nonmixing case , 2003, math/0304056.

[6]  J. Crutchfield,et al.  Regularities unseen, randomness observed: levels of entropy convergence. , 2001, Chaos.

[7]  A. Makowski,et al.  Discrete-time filtering for linear systems with non-Gaussian initial conditions: asymptotic behavior of the difference between the MMSE and LMSE estimates , 1992 .

[8]  Sean R. Eddy,et al.  Profile hidden Markov models , 1998, Bioinform..

[9]  Gersende Fort,et al.  Forgetting the initial distribution for Hidden Markov Models , 2007 .

[10]  Predrag R. Jelenkovic,et al.  State Learning and Mixing in Entropy of Hidden Markov Processes and the Gilbert-Elliott Channel , 1999, IEEE Trans. Inf. Theory.

[11]  Lalit R. Bahl,et al.  A Maximum Likelihood Approach to Continuous Speech Recognition , 1983, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[12]  David Haussler,et al.  Combining phylogenetic and hidden Markov models in biosequence analysis , 2003, RECOMB '03.

[13]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[14]  James P. Crutchfield,et al.  Exact Synchronization for Finite-State Sources , 2010, ArXiv.

[15]  D. Blackwell,et al.  On the Identifiability Problem for Functions of Finite Markov Chains , 1957 .

[16]  James P. Crutchfield,et al.  Asymptotic Synchronization for Finite-State Sources , 2010, ArXiv.

[17]  Sean R. Eddy,et al.  Maximum Discrimination Hidden Markov Models of Sequence Consensus , 1995, J. Comput. Biol..

[18]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[19]  M. A. McClure,et al.  Hidden Markov models of biological primary sequence information. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

[20]  R. Atar,et al.  Exponential stability for nonlinear filtering , 1997 .

[21]  Richard Hughey,et al.  Hidden Markov models for detecting remote protein homologies , 1998, Bioinform..

[22]  L. Baum,et al.  Statistical Inference for Probabilistic Functions of Finite State Markov Chains , 1966 .

[23]  Paul H. Siegel,et al.  On the capacity of finite state channels and the analysis of convolutional accumulate-m codes , 2003 .

[24]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[25]  F. Leonardi,et al.  Loss of memory of random functions of Markov chains and Lyapunov exponents , 2009, 0908.0077.

[26]  Haim H. Permuter,et al.  On the capacity of finite-state channels , 2008, 2008 IEEE International Symposium on Information Theory.

[27]  C. SIAMJ..  LYAPUNOV EXPONENTS FOR FINITE STATE NONLINEAR FILTERING , 1997 .

[28]  Brian H. Marcus,et al.  Analyticity of Entropy Rate of Hidden Markov Chains , 2005, IEEE Transactions on Information Theory.

[29]  F. Jelinek,et al.  Continuous speech recognition by statistical methods , 1976, Proceedings of the IEEE.

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

[31]  Jaroslav Kožešnk,et al.  Information Theory, Statistical Decision Functions, Random Processes , 1962 .

[32]  Laurent Mevel,et al.  Exponential Forgetting and Geometric Ergodicity in Hidden Markov Models , 2000, Math. Control. Signals Syst..