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

arXiv: https://arxiv.org/abs/1901.03732 · HTML: https://mlsanity.com/paper/1901.03732/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jan 2019.

## Built on by (most cited first, 1 of 1)

- [Fast approximations of the Jeffreys divergence between univariate Gaussian mixture models via exponential polynomial densities](https://mlsanity.com/paper/2107.05901/) (2021)

## Abstract (arXiv)

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.