Generalizing the Fano Inequality

The Fano inequality gives a lower bound on the mutual information between two random variables that take values on an M-element set, provided at least one of the random variables is equiprobable. We show several simple lower bounds on mutual information which do not assume such a restriction. In particular, this can be accomplished by replacing log M with the infinite-order Renyi entropy in the Fano inequality. Applications to hypothesis testing are exhibited along with bounds on mutual information in terms of the a priori and a posteriori error probabilities.