Optimal measures and Markov transition kernels

Belavkin, Roman V. ORCID logoORCID: https://orcid.org/0000-0002-2356-1447 (2012) Optimal measures and Markov transition kernels. Journal of Global Optimization, 55 (2) . pp. 387-416. ISSN 0925-5001 [Article] (doi:10.1007/s10898-012-9851-1)

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Abstract

We study optimal solutions to an abstract optimization problem for measures, which is a generalization of classical variational problems in information theory and statistical physics. In the classical problems, information and relative entropy are defined using the Kullback-Leibler divergence, and for this reason optimal measures belong to a one-parameter exponential family. Measures within such a family have the property of mutual absolute continuity. Here we show that this property characterizes other families of optimal positive measures if a functional representing information has a strictly convex dual. Mutual absolute continuity of optimal probability measures allows us to strictly separate deterministic and non-deterministic Markov transition kernels, which play an important role in theories of decisions, estimation, control, communication and computation. We show that deterministic transitions are strictly sub-optimal, unless information resource with a strictly convex dual is unconstrained. For illustration, we construct an example where, unlike non-deterministic, any deterministic kernel either has negatively infinite expected utility (unbounded expected error) or communicates infinite information.

Item Type: Article
Research Areas: A. > School of Science and Technology > Computer Science
A. > School of Science and Technology > Computer Science > Artificial Intelligence group
Item ID: 8531
Notes on copyright: The final publication is available at www.springerlink.com
Useful Links:
Depositing User: Dr Roman Belavkin
Date Deposited: 01 Mar 2012 05:30
Last Modified: 06 Jun 2022 16:33
URI: https://eprints.mdx.ac.uk/id/eprint/8531

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