Entropy: The Markov Ordering Approach

Entropy

Alexander N. Gorban¹  Pavel A. Gorban¹ 

[1] ¹Department of Mathematics, University of Leicester, Leicester, UK ²Institute of Space and Information Technologies, Siberian Federal University, Krasnoyarsk, Russia ³Department of Resource Economics, University of California, Berkeley, CA, USA

关键词: Markov process;    Lyapunov function;    entropy functionals;    attainable region;    MaxEnt;    inference;   

DOI  :  10.3390/e12051145

来源: mdpi

【 摘 要 】

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 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.

【 授权许可】

CC BY   
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.