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Entropy of Hidden Markov Processes and Connections to Dynamical Systems Published online by Cambridge University Press: 05 June 2011
Abstract. Starting from the full shift on a finite alphabet A, by mingling some symbols of A, we obtain a new full shift on a smaller alphabet B. This amalgamation defines a factor map from (Aℕ,TA) to (Bℕ, TB), where TA and TB are the respective shift maps. According to the thermodynamic formalism, to each regular function (“potential”) ψ:Aℕ → ℝ, we can associate a unique Gibbs measure µψ. In this article, we prove that, for a large class of potentials, the pushforward measure µψ ∘ π−1 is still Gibbsian for a potential φ:Bℕ→ℝ having a “bit less” regularity than ψ. In the special case where ψ is a “two-symbol” potential, the Gibbs measure µψ is nothing but a Markov measure and the amalgamation π defines a hidden Markov chain. In this particular case, our theorem can be recast by saying that a hidden Markov chain is a Gibbs measure (for a Hölder potential).
Introduction
From different viewpoints and under different names, the so-called hidden Markov measures have received a lot of attention in the last fifty years [3]. One considers a (stationary) Markov chain (Xn)n∈ℕ with finite state space A and looks at its “instantaneous” image Yn ≔ π(Xn), where the map π is an amalgamation of the elements of A yielding a smaller state space, say B. It is well known that in general the resulting chain, (Yn)n∈ℕ, has infinite memory.
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