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# Approximations for the Entropy for Functions of Markov Chains

John J. Birch
The Annals of Mathematical Statistics
Vol. 33, No. 3 (Sep., 1962), pp. 930-938
Stable URL: http://www.jstor.org/stable/2237870
Page Count: 9
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## Abstract

If {Yn} is a stationary ergodic Markov process taking on values in a finite set {1, 2, ⋯, A}, then its entropy can be calculated directly. If φ is a function defined on 1, 2, ⋯, A, with values 1, 2, ⋯, D, no comparable formula is available for the entropy of the process {Yn = φ(Yn)}. However, the entropy of this functional process can be approximated by the monotonic functions Ḡn = h(Xn ∣ Xn-1, ⋯, X1) and $\underline{G}_n = h(X_n \mid X_{n-1}, \cdots, X_1, Y_0)$, the conditional entropies. Furthermore, if the underlying Markov process {Yn} has strictly positive transition probabilities, these two approximations converge exponentially to the entropy H, where the convergence is given by 0 ≤ Ḡn - H ≤ Bρ n-1 and $0 \leqq H - \underline{G}_n \leqq B\rho^{n-1}$ with $0 < \rho < 1, \rho$ being independent of the function φ.

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