Entropy is the Only Finitely Observable Invariant

Donald Ornstein () and Benjamin Weiss ()

Discussion Paper Series from The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem

Abstract: Our main purpose is to present a very surprising new characterization of the Shannon entropy of stationary ergodic processes. We will use two basic concepts: isomorphism of stationary processes and a notion of finite observability, and we will see how one is led, inevitably, to Shannon's entropy. A function J with values in some metric space, defined on all finite-valued, stationary, ergodic processes is said to be finitely observable (FO) if there is a sequence of functions S_{n}(x_{1},x_{2},...,x_{n}) that for all processes X converges to J(X) for almost every realization x_{1}^{\infty} of X. It is called an invariant if it returns the same value for isomorphic processes. We show that any finitely observable invariant is necessarily a continuous function of the entropy. Several extensions of this result will also be given.

Pages: 15 pages
Date: 2006-05
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