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Optimal measures and Markov transition kernels

Listed author(s):
  • Roman Belavkin

    ()

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    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

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    File URL: http://hdl.handle.net/10.1007/s10898-012-9851-1
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    Article provided by Springer in its journal Journal of Global Optimization.

    Volume (Year): 55 (2013)
    Issue (Month): 2 (February)
    Pages: 387-416

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    Handle: RePEc:spr:jglopt:v:55:y:2013:i:2:p:387-416
    DOI: 10.1007/s10898-012-9851-1
    Contact details of provider: Web page: http://www.springer.com

    Order Information: Web: http://www.springer.com/business/operations+research/journal/10898

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