Optimal measures and Markov transition kernels
Article
Belavkin, R. 2013. Optimal measures and Markov transition kernels. Journal of Global Optimization. 55 (2), pp. 387-416. https://doi.org/10.1007/s10898-012-9851-1
| Type | Article |
|---|---|
| Title | Optimal measures and Markov transition kernels |
| Authors | Belavkin, R. |
| 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. |
| Keywords | Expected utility; Information distance; Optimal policy; Radon measure; Randomized algorithm |
| Sustainable Development Goals | 9 Industry, innovation and infrastructure |
| Middlesex University Theme | Sustainability |
| Research Group | Artificial Intelligence group |
| Publisher | Springer |
| Journal | Journal of Global Optimization |
| ISSN | 0925-5001 |
| Electronic | 1573-2916 |
| Publication dates | |
| Online | 08 Feb 2012 |
| Feb 2013 | |
| Publication process dates | |
| Deposited | 01 Mar 2012 |
| Accepted | 14 Jan 2012 |
| Submitted | 01 Dec 2010 |
| Output status | Published |
| Copyright Statement | The final publication is available at www.springerlink.com: http://dx.doi.org/10.1007/s10898-012-9851-1 |
| Digital Object Identifier (DOI) | https://doi.org/10.1007/s10898-012-9851-1 |
| Scopus EID | 2-s2.0-84878576380 |
| Web of Science identifier | WOS:000314278100011 |
| Language | English |
| File |
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