最优测度与马尔可夫转移核
Optimal measures and Markov transition kernels
摘要：
研究测度的抽象最优化问题的最优解，这是信息论和统计物理中经典变分问题的推广.在经典问题中，信息和相对熵是使用Kullback-Leibler分歧定义的，因此最佳度量属于单参数指数族。这样一个族中的测度具有相互绝对连续的性质。在这里，我们表明，这一性质的特点是其他家庭的最佳积极措施，如果一个功能表示的信息有一个严格凸的对偶。最优概率测度的相互绝对连续性使我们能够严格区分确定性和非确定性马尔可夫转移核，它们在决策、估计、控制、通信和计算理论中起着重要作用。我们表明，确定性的过渡是严格的次优，除非信息资源与严格凸的对偶是无约束的。为了说明，我们构造了一个例子，与非确定性不同，任何确定性内核都具有负无限的预期效用（无限的预期误差）或传递无限的信息。
英文摘要：
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.
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