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

Alexander Mielke

TL;DR

The work provides a cohesive, expository treatment of the Hellinger distance and the induced Fisher-Rao framework for subsets of measures, linking historical origins to modern geometric and dynamical viewpoints. It introduces a complete dynamic characterization of absolutely continuous curves in the Hellinger space via a growth equation, and derives explicit geodesic formulas and embedding tools that reveal a flat, Hilbert-space-like structure. A general Fisher-Rao construction on arbitrary subsets is developed, including the Bhattacharya distance for probability measures and the cone decomposition, with detailed analysis of product measures. The authors then illustrate the theory through classical distribution families—translations, Poisson, exponential, and Gaussian—deriving explicit distance metrics and discussing invariance properties, which clarifies how these metrics interact with transformations and product structures. Overall, the paper provides a solid theoretical foundation for gradient-flow analyses in Hellinger spaces and practical, computable formulas for key distribution families, with implications for information geometry and statistical inference.

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.

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

TL;DR

The work provides a cohesive, expository treatment of the Hellinger distance and the induced Fisher-Rao framework for subsets of measures, linking historical origins to modern geometric and dynamical viewpoints. It introduces a complete dynamic characterization of absolutely continuous curves in the Hellinger space via a growth equation, and derives explicit geodesic formulas and embedding tools that reveal a flat, Hilbert-space-like structure. A general Fisher-Rao construction on arbitrary subsets is developed, including the Bhattacharya distance for probability measures and the cone decomposition, with detailed analysis of product measures. The authors then illustrate the theory through classical distribution families—translations, Poisson, exponential, and Gaussian—deriving explicit distance metrics and discussing invariance properties, which clarifies how these metrics interact with transformations and product structures. Overall, the paper provides a solid theoretical foundation for gradient-flow analyses in Hellinger spaces and practical, computable formulas for key distribution families, with implications for information geometry and statistical inference.

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.

Paper Structure

This paper contains 20 sections, 5 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Properties of the Hellinger distance
  3. Hellinger's integral
  4. The topology of $(\mathfrak M(\Omega),\mathsf{He})$
  5. Absolutely continuous curves, metric speed and the continuity equation in $(\mathfrak M(\Omega),\mathsf{He})$
  6. Geodesics curves
  7. Properties similar to Hilbert-space geometry
  8. Hellinger distance for product measures
  9. Invariance under pushforwards of the Hellinger distance
  10. Historical remarks
  11. Various Fisher-Rao distances
  12. The general construction for subsets ${\mathcal{S}} \subset \mathfrak M(\Omega)$
  13. Bhattacharya distance alias spherical Hellinger distance
  14. General cones
  15. Product measures
  16. ...and 5 more sections

Key Result

Theorem 2.2

(A) If $\mu:[s_0,s_1]\to \mathfrak M(\Omega)$ is $2$-absolutely continuous in $(\mathfrak M(\Omega),\mathsf{He})$, then there exists $\xi \in {\mathrm L}^2([s_0,s_1]{\times} \Omega)$ such that $(\mu,\xi)$ solve the growth equation eq:GrowthEqn and and the metric speed satisfies (B) Vice versa, if $\mu :[s_0,s_1]\to \mathfrak M(\Omega)$ is a continuous curve, $\xi\in {\mathrm L}^2([s_0,s_1]{\times

Figures (1)

  • Figure 2.1: A visualization of the parallelogram identity in $(\mathfrak M(\Omega),\mathsf{He})$ where all edges (full lines) are geodesics, while the broken lines contain curves that may lead to signed measures lying in ${\mathrm L}^2(\Omega,\lambda)$. The center of the parallelogram is $\mu_A=A^\mathsf{He}(\mu_1,\mu_2)$.

Theorems & Definitions (9)

  • Remark 2.1: Kolmogorov and Hellinger integrals
  • Theorem 2.2
  • Remark 2.3: Embedding into Euclidean space
  • Lemma 2.4: Hellinger distance and pushforward
  • Remark 3.1: Unique characterization
  • Theorem 3.2: Fisher-Rao distance on cones
  • Proposition 3.3: Product probability measures
  • Remark 4.1
  • Theorem 4.2: Fisher-Rao distance within Gaussians