The Entropy of Conditional Markov Trajectories

TL;DR

This paper introduces a method to compute the entropy of Markov trajectories conditioned on intermediate states by transforming the original chain, enabling more precise analysis of trajectory randomness.

Contribution

It proposes a novel transformation technique for Markov chains to calculate the entropy of conditioned trajectories and expresses global entropy as a sum of local state entropies.

Findings

Provides a method for computing conditional trajectory entropy

Expresses global entropy as a linear combination of local entropies

Enables analysis of trajectory randomness with intermediate states

Abstract

To quantify the randomness of Markov trajectories with fixed initial and final states, Ekroot and Cover proposed a closed-form expression for the entropy of trajectories of an irreducible finite state Markov chain. Numerous applications, including the study of random walks on graphs, require the computation of the entropy of Markov trajectories conditioned on a set of intermediate states. However, the expression of Ekroot and Cover does not allow for computing this quantity. In this paper, we propose a method to compute the entropy of conditional Markov trajectories through a transformation of the original Markov chain into a Markov chain that exhibits the desired conditional distribution of trajectories. Moreover, we express the entropy of Markov trajectories - a global quantity - as a linear combination of local entropies associated with the Markov chain states.