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Optimal measures and transition kernels.

Ср, 12/15/2010 - 16:01 — SomNambul
Authors: Roman V. Belavkin
Subjects: Optimization and Control
link: http://arxiv.org/abs/1012.0366
Abstract

We study positive measures that are solutions to an abstract optimisation
problem, which is a generalisation of a classical variational problem with a
constraint on information of a Kullback-Leibler type. The latter leads to
solutions that belong to a one parameter exponential family, and such measures
have the property of mutual absolutely continuity. Here we show that this
property is related to strict convexity of a functional that is dual to the
functional representing information, and therefore mutual absolute continuity
characterises other families of optimal measures. This result plays an
important role in problems of optimal transitions between two sets: Mutual
absolute continuity implies that optimal transition kernels cannot be
deterministic, unless information is unbounded. 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.

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