抽象空间上的有向信息：性质与变分等式
Directed Information on Abstract Spaces: Properties and Variational Equalities
摘要：
有向信息或其变体被广泛用于表征具有记忆和反馈的信道的容量、非预期的有损数据压缩以及它们对网络的推广。本文利用概率测度的弱收敛拓扑，得到了一般抽象字母表（完备可分度量空间）的有向信息的几个泛函和拓扑性质。这些问题包括一致分布集的凸性，一致分布集唯一定义了因果条件分布，一致分布集的有向信息的凸性和非凸性，这些分布集的弱紧性，它们的联合分布及其边缘。此外，我们还证明了有向信息的下连续性，并在一定条件下建立了有向信息的连续性。最后，我们推导出变分等式的有向信息，包括顺序版本。这些可以被看作是互信息的变分等式（用于Blahut-Arimoto算法）的模拟。总之，我们将互信息的基本泛函和拓扑性质推广到有向信息。这些属性的背景下讨论的极值问题的有向信息。
英文摘要：
Directed information or its variants are utilized extensively in the characterization of the capacity of channels with memory and feedback, nonanticipative lossy data compression, and their generalizations to networks. In this paper, we derive several functional and topological properties of directed information for general abstract alphabets (complete separable metric spaces) using the topology of weak convergence of probability measures. These include convexity of the set of consistent distributions, which uniquely define causally conditioned distributions, convexity and concavity of directed information with respect to the sets of consistent distributions, weak compactness of these sets of distributions, their joint distributions and their marginals. Furthermore, we show lower semicontinuity of directed information, and under certain conditions we also establish continuity of directed information. Finally, we derive variational equalities for directed information, including sequential versions. These may be viewed as the analogue of the variational equalities of mutual information (utilized in Blahut-Arimoto algorithm). In summary, we extend the basic functional and topological properties of mutual information to directed information. These properties are discussed in the context of extremum problems of directed information.
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