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Lumpings of Markov Chains, Entropy Rate Preservation, and Higher-Order Lumpability

Part of: Probabilistic methods, simulation and stochastic differential equations Communication, information Markov processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Bernhard C. Geiger  and
Christoph Temmel
Bernhard C. Geiger*
Affiliation:
Graz University of Technology
Christoph Temmel*
Affiliation:
Graz University of Technology and VU University Amsterdam
*
∗ Postal address: Institute for Communications Engineering, Technische Universiät München, Theresienstrasse 90, 80333 Munich. Email address: geiger@ieee.org
∗∗ Postal address: Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address: math@temmel.me
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Abstract

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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.

Keywords

Lumpingentropy rate lossfunctional hidden Markov modelstrong lumpabilityhigher-order Markov chain

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60G17: Sample path properties 94A17: Measures of information, entropy 60G10: Stationary processes 65C40: Computational Markov chains
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1114 - 1132
Copyright
© Applied Probability Trust 

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