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

## TL;DR

This paper introduces new statistical Minkowski distances with closed-form formulas for Gaussian mixtures and other exponential family mixtures, enabling efficient computation of distances and diversity indices in probabilistic models.

## Contribution

It extends Minkowski distances to probability densities, providing closed-form formulas for Gaussian mixtures and other exponential family mixtures, which was not previously available.

## Key findings

- Closed-form formulas for Gaussian mixture distances.
- Extension to mixtures of exponential families.
- Introduction of Minkowski's diversity index.

## Abstract

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. 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.