Optimizing Bivariate Partial Information Decomposition

None of the BROJA information decomposition measures $\mbox{SI}, \mbox{CI}, \mbox{UIy}, \mbox{UIz}$ are convex or concave over the probability simplex. In this paper, we provide formulas for the sub-gradient and super-gradients of any of the information decomposition measures. Then we apply these results to obtain an optimum of some of these information decomposition measures when optimized over a constrained set of probability distributions.