Level Sets Based Distances for Probability Measures and Ensembles with Applications

TL;DR

This paper introduces a novel family of distances for probability measures by viewing them as functionals in a functional space, utilizing density level set estimation for practical measurement.

Contribution

It proposes a new non-parametric metric for probability measures based on their action on functions and density level set estimation, expanding the tools for statistical analysis.

Findings

The new distance performs well compared to existing metrics.

It effectively captures differences between probability measures.

Applications on real and simulated data demonstrate its utility.

Abstract

In this paper we study Probability Measures (PM) from a functional point of view: we show that PMs can be considered as functionals (generalized functions) that belong to some functional space endowed with an inner product. This approach allows us to introduce a new family of distances for PMs, based on the action of the PM functionals on `interesting' functions of the sample. We propose a specific (non parametric) metric for PMs belonging to this class, based on the estimation of density level sets. Some real and simulated data sets are used to measure the performance of the proposed distance against a battery of distances widely used in Statistics and related areas.