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Investigating Directed Information

Final Report Summary - DIRECTEDINFO (Investigating Directed Information)

This research investigates a new measure that arises in information theory called directed information. Our research show that directed information arises in communication as the maximum rate (a.k.a. capacity) that can be transmitted reliably in channels with feedback. Beyond communication we have showed that directed information is a measure of causality between two processes. In addition it characterizes the value of causal side information in the growth of a gambling or portfolio investment problem as well as the work extracted from causal side information from knowledge in statistical physics.

The directed information is multi-letter expression (optimization over an infinite sum) and therefore very difficult to optimize or compute. Hence, we invented a new idea of graph-based auxiliary random variable that allows us to transform the multi-letter expression into a single letter convex optimization problem. Furthermore, in order to identify the structure of the graph-based auxiliary random variable we have introduced a reinforcement learning problem that its solution yields the desired structure. Right now these two novels ideas works well for finding the fundamental limits (capacity) of large families of finite state channel with feedback and we hope to develop these ideas for all problems where directed information plays an important role in communication and beyond.