Canalizing Boolean Functions Maximize Mutual Information

Abstract

Information processing in biologically motivated Boolean networks is of interest in recent information theoretic research. One measure to quantify this ability is the well-known mutual information. Using Fourier analysis, we show that canalizing functions maximize mutual information between a single input variable and the outcome of a function with fixed expectation. A similar result can be obtained for the mutual information between a set of input variables and the output. Further, if the expectation of the function is not fixed, we obtain that the mutual information is maximized by a function only dependent on this single variable, i.e., the dictatorship function. We prove our findings for Boolean functions with uniformly distributed as well as product distributed input variables.