Journal of Applied Probability

Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability

Bernhard C. Geiger and Christoph Temmel

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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.

Article information

J. Appl. Probab. Volume 51, Number 4 (2014), 1114-1132.

First available in Project Euclid: 20 January 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Lumping entropy rate loss functional hidden Markov model strong lumpability higher-order Markov chain


Geiger, Bernhard C.; Temmel, Christoph. Lumpings of Markov chains, entropy rate preservation, and higher-order lumpability. J. Appl. Probab. 51 (2014), no. 4, 1114--1132.

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