This research investigates a new measure that arises in information theorycalled directed information. Recent advances, including our preliminary results, shows thatdirected information arises in communication as the maximum rate that can be transmitted reliablyin channels with feedback. The directed information is multi-letter expression and therefore veryhard to optimize or compute.Our plan is first of all to find an efficient methodology for optimizing the measure using thedynamic programming framework and convex optimization tools. As an important by-product offinding the fundamental limits is finding coding schemes that achieves the limits. Second, weplan to find new roles for directed information in communication, especially in networks withbi-directional communication and in data compression with causal conditions. Third, encouraged bya preliminary work on interpretation of directed information in economics and estimation theory,we plan to show that directed information has interpretation in additional fields such asstatistical physics. We plan to show that there is duality relation between different fields withcausal constraints. Due to the duality insights and breakthroughs in one problem will lead to newinsights in other problems. Finally, we will apply directed information as a statisticalinference of causal dependence. We will show how to estimate and use the directed informationestimator to measure causal inference between two or more process. In particular, one of thequestions we plan to answer is the influence of industrial activities (e.g., $\text{CO}_2$volumes) on the global warming.Our main focus will be to develop a deeper understanding of the mathematical properties ofdirected information, a process that is instrumental to each problem. Due to their theoreticalproximity and their interdisciplinary nature, progress in one problem will lead to new insightsin other problems. A common set of mathematical tools developed in