Translator Disclaimer
December 2014 Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability
Bernhard C. Geiger, Christoph Temmel
Author Affiliations +
J. Appl. Probab. 51(4): 1114-1132 (December 2014).

Abstract

A lumping of a Markov chain is a coordinatewise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, if and only if the lumped process is a kth-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

Citation

Download Citation

Bernhard C. Geiger. Christoph Temmel. "Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability." J. Appl. Probab. 51 (4) 1114 - 1132, December 2014.

Information

Published: December 2014
First available in Project Euclid: 20 January 2015

zbMATH: 1309.60077
MathSciNet: MR3301292

Subjects:
Primary: 60J10
Secondary: 60G10, 60G17, 65C40, 94A17

Rights: Copyright © 2014 Applied Probability Trust

JOURNAL ARTICLE
19 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.51 • No. 4 • December 2014
Back to Top