Entropy: The Markov Ordering Approach

Details Entropy: The Markov Ordering Approach Entropy: The Markov Ordering Approach

2010-03-06

arxiv.org/abs/1003.1377v4

Entropy. 2010; 12(5):1145-1193

Physics

Data Analysis, Statistics and Probability

49 pages, 4 figures, Journal (open access) version with minor corrections in terminology

Scientific paper

10.3390/e12051145

The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions.

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