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Directed information or its variants are utilized extensively in thecharacterization 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 ofdirected information for general abstract alphabets (complete separable metricspaces) using the topology of weak convergence of probability measures. Theseinclude convexity of the set of consistent distributions, which uniquely definecausally conditioned distributions, convexity and concavity of directedinformation with respect to the sets of consistent distributions, weakcompactness of these sets of distributions, their joint distributions and theirmarginals. Furthermore, we show lower semicontinuity of directed information,and under certain conditions we also establish continuity of directedinformation. Finally, we derive variational equalities for directedinformation, including sequential versions. These may be viewed as the analogueof the variational equalities of mutual information (utilized in Blahut-Arimotoalgorithm).  In summary, we extend the basic functional and topological properties ofmutual information to directed information. These properties are discussed inthe context of extremum problems of directed information.

Directed Information on Abstract Spaces: Properties and Variational Equalities