Optimal measures and Markov transition kernels
Belavkin, Roman V. (2012) Optimal measures and Markov transition kernels. Journal of Global Optimization, 55 (2). pp. 387-416. ISSN 0925-5001
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Official URL: http://dx.doi.org/10.1007/s10898-012-9851-1
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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.
|Research Areas:||Science & Technology > Science & Technology|
Science & Technology > Algorithms, Architecture & Networks
Science & Technology > Mathematics & Statistics
|Permissions granted by publisher:||The final publication is available at www.springerlink.com|
|Deposited On:||01 Mar 2012 05:30|
|Last Modified:||31 May 2013 14:03|
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