Optimal measures and transition kernels
Title: Optimal measures and transition kernels
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.