Expected time averages in Markovian imprecise jump processes: a graph-theoretic characterisation of weak ergodicity

Yema Paul, Alexander Erreygers, Jasper De Bock
Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, PMLR 215:379-389, 2023.

Abstract

Markovian imprecise jump processes provide a way to express model uncertainty about Markovian jump processes. The dynamics are not governed by a unique rate matrix, but are instead partially specified by a set of such matrices. Since the dynamics are partially specified, the resulting expected time averages are no longer uniquely determined either, and one then resorts to tight lower and upper bounds on them. In this paper, we are interested in the existence of an asymptotic limit of these upper and lower bounds, as the time horizon becomes infinite. When those limits exist and are furthermore independent of the choice of the process’s initial state, we say that the process is weakly ergodic. Our main contribution is a necessary and sufficient condition for a Markovian imprecise jump process to be weakly ergodic, expressed in terms of simple graph-theoretic conditions on its set of rate matrices.

Cite this Paper

BibTeX
@InProceedings{pmlr-v215-paul23a, title = {Expected time averages in {M}arkovian imprecise jump processes: a graph-theoretic characterisation of weak ergodicity}, author = {Paul, Yema and Erreygers, Alexander and De Bock, Jasper}, booktitle = {Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications}, pages = {379--389}, year = {2023}, editor = {Miranda, Enrique and Montes, Ignacio and Quaeghebeur, Erik and Vantaggi, Barbara}, volume = {215}, series = {Proceedings of Machine Learning Research}, month = {11--14 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v215/paul23a/paul23a.pdf}, url = {https://proceedings.mlr.press/v215/paul23a.html}, abstract = {Markovian imprecise jump processes provide a way to express model uncertainty about Markovian jump processes. The dynamics are not governed by a unique rate matrix, but are instead partially specified by a set of such matrices. Since the dynamics are partially specified, the resulting expected time averages are no longer uniquely determined either, and one then resorts to tight lower and upper bounds on them. In this paper, we are interested in the existence of an asymptotic limit of these upper and lower bounds, as the time horizon becomes infinite. When those limits exist and are furthermore independent of the choice of the process’s initial state, we say that the process is weakly ergodic. Our main contribution is a necessary and sufficient condition for a Markovian imprecise jump process to be weakly ergodic, expressed in terms of simple graph-theoretic conditions on its set of rate matrices.} }
Endnote
%0 Conference Paper %T Expected time averages in Markovian imprecise jump processes: a graph-theoretic characterisation of weak ergodicity %A Yema Paul %A Alexander Erreygers %A Jasper De Bock %B Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2023 %E Enrique Miranda %E Ignacio Montes %E Erik Quaeghebeur %E Barbara Vantaggi %F pmlr-v215-paul23a %I PMLR %P 379--389 %U https://proceedings.mlr.press/v215/paul23a.html %V 215 %X Markovian imprecise jump processes provide a way to express model uncertainty about Markovian jump processes. The dynamics are not governed by a unique rate matrix, but are instead partially specified by a set of such matrices. Since the dynamics are partially specified, the resulting expected time averages are no longer uniquely determined either, and one then resorts to tight lower and upper bounds on them. In this paper, we are interested in the existence of an asymptotic limit of these upper and lower bounds, as the time horizon becomes infinite. When those limits exist and are furthermore independent of the choice of the process’s initial state, we say that the process is weakly ergodic. Our main contribution is a necessary and sufficient condition for a Markovian imprecise jump process to be weakly ergodic, expressed in terms of simple graph-theoretic conditions on its set of rate matrices.
APA
Paul, Y., Erreygers, A. & De Bock, J.. (2023). Expected time averages in Markovian imprecise jump processes: a graph-theoretic characterisation of weak ergodicity. Proceedings of the Thirteenth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 215:379-389 Available from https://proceedings.mlr.press/v215/paul23a.html.

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