Generalization of the Energy Distance by Bernstein Functions

Abstract

We reprove the well-known fact that the energy distance defines a metric on the space of Borel probability measures on a Hilbert space with finite first moment by a new approach, by analyzing the behavior of the Gaussian kernel on Hilbert spaces and a maximum mean discrepancy analysis. From this new point of view, we are able to generalize the energy distance metric to a family of kernels related to Bernstein functions and conditionally negative definite kernels. We also explain what occurs on the energy distance on the kernel $$\Vert x-y\Vert ^{a}$$ for every $$a >2$$ , by describing in which circumstances it defines a distance between probabilities. We also generalize this idea to a family of kernels related to completely monotone functions of finite order and conditionally negative definite kernels.