The statistical Minkowski distances: Closed-form formula for Gaussian Mixture Models

The traditional Minkowski distances are induced by the corresponding Minkowski norms in real-valued vector spaces. In this work, we propose novel statistical symmetric distances based on the Minkowski's inequality for probability densities belonging to Lebesgue spaces. These statistical Minkowski distances admit closed-form formula for Gaussian mixture models when parameterized by integer exponents: Namely, we prove that these distances between mixtures are obtained from multinomial expansions, and written by means of weighted sums of inverse exponentials of generalized Jensen diversity indices of the mixture component distributions. This result extends to arbitrary mixtures of exponential families with natural parameter spaces being cones: This includes the binomial, the multinomial, the zero-centered Laplacian, the Gaussian and the Wishart mixtures, among others. We also derive a Minkowski's diversity index of a normalized weighted set of probability distributions from Minkowski's inequality.