Optimal measures and Markov transition kernels
arXiv:1012.0366 · doi:10.1007/s10898-012-9851-1
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.
Replaced with a final and accepted draft; Journal of Global Optimization, Springer, Jan 1, 2012
References in corpus (2)
Cited by in corpus (7)
- Swarm Intelligence Based Algorithms: A Critical Analysis
- Analysis of Agent Expertise in Ms. Pac-Man using Value-of-Information-based Policies
- Monotonicity of Fitness Landscapes and Mutation Rate Control
- Relation between the Kantorovich-Wasserstein metric and the Kullback-Leibler divergence
- The algebra and machine representation of statistical models
- Asymmetry of Risk and Value of Information
- Value of Information in the Mean-Square Case and its Application to the Analysis of Financial Time-Series Forecast