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Optimal measures and Markov transition kernels

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  • Roman Belavkin

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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. Copyright Springer Science+Business Media, LLC. 2013

Suggested Citation

  • Roman Belavkin, 2013. "Optimal measures and Markov transition kernels," Journal of Global Optimization, Springer, vol. 55(2), pages 387-416, February.
  • Handle: RePEc:spr:jglopt:v:55:y:2013:i:2:p:387-416
    DOI: 10.1007/s10898-012-9851-1
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    File URL: http://hdl.handle.net/10.1007/s10898-012-9851-1
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    Cited by:

    1. Roman V. Belavkin, 2014. "Asymmetry of Risk and Value of Information," SEET Working Papers 2014-03, BELIS, Istanbul Bilgi University.

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