A novel measure of multivariate redundant information

Consider a situation in which a set of n random variables X1...Xn carry information about some random variable of interest Y. For example, Y could represent an external stimulus while X1...Xn could represent the activity of n brain regions; alternatively, Y could be the output of a digital logic gate while X1...Xn could be the n inputs of the gate. Recent work in information theory has considered the problem of quantifying the amount of redundancy among X1...Xn about Y, i.e., the amount of information about Y that is found individually within all X1...Xn. Several approaches to quantifying redundancy have been proposed, but most lack a clear operational motivation, or only apply in the case of two X variables. Here we propose a novel measure of redundancy which applies to an arbitrary number of variables. We show that our measure has decision-theoretic, set-algebraic, geometric, and axiomatic interpretations. Our approach also sheds new light on the difficulties encountered by existing ways of quantifying redundant and synergistic information.