On Relationship between Mutual Information and Variation

Investigation of a relationship between the mutual information and variational distance, started in Pinsker paper [1], where an upper bound for the mutual information via variational distance was obtained, is here continued. We present a simple lower bound which is optimal or asymptotically optimal in some cases. An uniform upper bound for the mutual information via variational distance is also derived for random variables with a finite number of values. For such random variables, the asymptotic behaviour of the maximum of mutual information is also investigated in the case where the variational distance tends to zero or to its maximum value.