WIAS Preprint No. 3222, (2025)

Some notes on the Hellinger distance and various Fisher--Rao distances


Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2020 Mathematics Subject Classification

  • 46G99 01A60 53C22 58E10 94A17

Keywords

  • Hellinger distance, Fisher-Rao distance, Bhattacharya distance, geodesic curves, product measures, Gaussian measures, exponential distributions, Poisson distributions, metric cone, absolutely continuous curves, growth equation

DOI

10.20347/WIAS.PREPRINT.3222

Abstract

These expository notes introduce the Hellinger distance on the set of all measures and the induced Fisher-Rao distances for subsets of measures, such as probability measures or Gaussian measures. The historical background is highlighted and the relations and the distinct features of the two distances are discussed. Moreover, we provide a dynamic characterization of absolutely continuous curves in the Hellinger spaces in terms of the growth equation, which replaces the continuity equation in the theory of optimal transport.

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