Entropy: The Markov Ordering Approach

TL;DR

This paper introduces the Markov order based on entropy and invariance properties, providing a new way to compare distributions and analyze Markov processes beyond traditional entropy measures.

Contribution

It defines the Markov order using invariance and additivity properties, offering a novel framework for understanding Markov processes and distribution comparison.

Findings

Derived all Lyapunov functionals with specified invariance properties.

Described the most general Markov order for distribution comparison.

Showed differences between Markov order and entropy growth ordering.

Abstract

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.