Revisiting Chernoff Information with Likelihood Ratio Exponential Families

The Chernoff information between two probability measures is a statistical divergence measuring their distinguishability defined as their maximally skewed Bhattacharyya distance. Although the Chernoff information was originally defined for bounding the Bayes error in statistical hypothesis testing, the divergence found many other applications ranging from information fusion to quantum information theory. From the viewpoint of information theory, the Chernoff information can also be interpreted as a minimax symmetrization of the Kullback--Leibler divergence related to the capacity of a discrete memoryless channel. In this paper, we first revisit the Chernoff information by considering the exponential families induced by geometric mixtures: The likelihood ratio exponential families. Second, we show how to (i) calculate using symbolic computing a closed-form formula for the Chernoff information between any two univariate Gaussian distributions and (ii) use a fast numerical scheme to approximate the Chernoff information between any two multivariate Gaussian distributions. We also report a closed-form formula of the Chernoff information of centered Gaussians with scaled covariance matrices.