  The Chernoff information between two probability measures is a statistical
divergence measuring their deviation defined as their maximally skewed
Bhattacharyya distance. Although the Chernoff information was originally
introduced for bounding the Bayes error in statistical hypothesis testing, the
divergence found many other applications due to its empirical robustness
property found in applications ranging from information fusion to quantum
information. From the viewpoint of information theory, the Chernoff information
can also be interpreted as a minmax symmetrization of the Kullback--Leibler
divergence. In this paper, we first revisit the Chernoff information between
two densities of a measurable Lebesgue space by considering the exponential
families induced by their geometric mixtures: The so-called likelihood ratio
exponential families. Second, we show how to (i) solve exactly the Chernoff
information between any two univariate Gaussian distributions or get a
closed-form formula using symbolic computing, (ii) report a closed-form formula
of the Chernoff information of centered Gaussians with scaled covariance
matrices and (iii) use a fast numerical scheme to approximate the Chernoff
information between any two multivariate Gaussian distributions.
