Analyticity of entropy rate in families of hidden markov chains

We prove that under mild assumptions a hidden Markov chain varies analytically, in a strong sense, as a function of the underlying Markov chain parameters. In particular, we show that, under these assumptions, the entropy rate of a hidden Markov chain is an analytic function of the parameters. We give examples to show how this can fail in some cases. And we study two natural special classes of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol and binary Markov chains corrupted by binary symmetric noise

[1]  Philippe Jacquet,et al.  On the entropy of a hidden Markov process , 2004, Data Compression Conference, 2004. Proceedings. DCC 2004.

[2]  Douglas Lind,et al.  An Introduction to Symbolic Dynamics and Coding , 1995 .

[3]  Venkat Anantharam,et al.  An upper bound for the largest Lyapunov exponent of a Markovian product of nonnegative matrices , 2005, Theor. Comput. Sci..

[4]  Eytan Domany,et al.  Asymptotics of the entropy rate for a hidden Markov process , 2005, Data Compression Conference.

[5]  Tsachy Weissman,et al.  New bounds on the entropy rate of hidden Markov processes , 2004, Information Theory Workshop.

[6]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[7]  D. Ruelle.  Differentiation of SRB States , 1997 .

[8]  Vladimir B. Balakirsky,et al.  On the entropy rate of a hidden Markov model , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[9]  John G. Kemeny,et al.  Finite Markov Chains. , 1960 .

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

[11]  P. Glynn,et al.  On Entropy and Lyapunov Exponents for Finite-State Channels , 2003 .

[12]  Örjan Stenflo,et al.  Uniqueness of Invariant Measures for Place-Dependent Random Iterations of Functions , 2002 .

[13]  Eytan Domany,et al.  Taylor series expansions for the entropy rate of Hidden Markov Processes , 2005, 2006 IEEE International Conference on Communications.

[14]  David Ruelle,et al.  Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics , 1978 .

[15]  D. Ruelle,et al.  Analycity properties of the characteristic exponents of random matrix products , 1979 .

[16]  Yuval Peres,et al.  Analytic dependence of Lyapunov exponents on transition probabilities , 1991 .

[17]  Tsachy Weissman,et al.  On the optimality of symbol-by-symbol filtering and denoising , 2004, IEEE Transactions on Information Theory.

[18]  Y. Peres.  Domains of analytic continuation for the top Lyapunov exponent , 1992 .

[19]  Joseph L. Taylor.  Several Complex Variables with Connections to Algebraic Geometry and Lie Groups , 2002 .

[20]  L. Arnold,et al.  Evolutionary Formalism for Products of Positive Random Matrices , 1994 .