Entropy of Hidden Markov Processes and Connections to Dynamical Systems BIRS 5-Day Workshop, 07w5103

I will present an overview on the entropy rate of hidden Markov processes (HMP’s), information and coding-theoretic motivation for its study, and some of its connections to dynamical systems, to non-linear filtering, and to statistical physics. Particular attention will be given to: – Alternative representations: via the Blackwell measure, as a Lyapunov exponent, and as a partition function in statistical physics. – Bounds and approximations (stochastic and deterministic), and their complexity-precision tradeoffs. – Asymptotic regimes and analyticity. • M. Boyle (Math, Maryland): Title: Overview of Markovian maps Abstract: A topological Markov shift is the support of a Markov chain (measure); that is, it is the set of infinite sequences all of whose finite subwords have strictly positive probability (measure). A topological Markov shift can support many different Markov chains, including higher-order chains (on which the past and future become independent after conditioning on finitely many steps in the past). Now let f be a sliding block code from a topological Markov shift S onto another topological Markov shift T. We assume S is irreducible (it is the support of an irreducible/ergodic Markov chain). Then there is a dichotomy: either every Markov measure on T lifts (via f) to a Markov measure on S, or every Markov measure on T does not lift to a Markov measure on S. In the former case, the map f is called Markovian. The Markovian condition is a thermodynamic phenomenon and is the first of a range of conditions on the regularity of the map f. I will try to explain this condition, the related conditions, and related work due to myself, Petersen, Quas, Shin, Tuncel, Walters and others. A topological Markov shift is the support of a Markov chain (measure); that is, it is the set of infinite sequences all of whose finite subwords have strictly positive probability (measure). A topological Markov shift can support many different Markov chains, including higher-order chains (on which the past and future become independent after conditioning on finitely many steps in the past). Now let f be a sliding block code from a topological Markov shift S onto another topological Markov shift T. We assume S is irreducible (it is the support of an irreducible/ergodic Markov chain). Then there is a dichotomy: either every Markov measure on T lifts (via f) to a Markov measure on S, or every Markov measure on T does not lift to a Markov measure on S. In the former case, the map f is called Markovian. The Markovian condition is a thermodynamic phenomenon and is the first of a range of conditions on the regularity of the map f. I will try to explain this condition, the related conditions, and related work due to myself, Petersen, Quas, Shin, Tuncel, Walters and others. • E. Verbitsky (Philips-Eindhoven): Title: Thermodynamics of Hidden Processes