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Relationship between Hölder Divergence and Functional Density Power Divergence: Intersection and Generalization

Masahiro Kobayashi

TL;DR

This paper investigates the relationship between Hölder divergence and the functional density power divergence (FDPD) within the broader landscape of density-power-based divergences. It establishes that the intersection of Hölder divergence and FDPD collapses to the Jones–Hölder–Héroux–Jones (JHHB) divergence family and introduces the $\xi$-Hölder divergence as a unifying generalization informed by Hölder’s inequality and composite scoring rules. The authors derive an inequality between composite scoring rules corresponding to different FDPDs under the $\xi$-Hölder framework and show that imposing the Hölder score structure on a composite scoring rule yields the $\xi$-Hölder divergence. The work clarifies how Hölder divergence and FDPD relate to known divergences (e.g., $DPD$, $\gamma$-divergence) as special cases and points to future extensions to negative $\gamma$ and non-composite divergences, with potential implications for robust statistical inference.

Abstract

In this study, we discuss the relationship between two families of density-power-based divergences with functional degrees of freedom -- the Hölder divergence and the functional density power divergence (FDPD) -- based on their intersection and generalization. These divergence families include the density power divergence and the $γ$-divergence as special cases. First, we prove that the intersection of the Hölder divergence and the FDPD is limited to a general divergence family introduced by Jones et al. (Biometrika, 2001). Subsequently, motivated by the fact that Hölder's inequality is used in the proofs of nonnegativity for both the Hölder divergence and the FDPD, we define a generalized divergence family, referred to as the $ξ$-Hölder divergence. The nonnegativity of the $ξ$-Hölder divergence is established through a combination of the inequalities used to prove the nonnegativity of the Hölder divergence and the FDPD. Furthermore, we derive an inequality between the composite scoring rules corresponding to different FDPDs based on the $ξ$-Hölder divergence. Finally, we prove that imposing the mathematical structure of the Hölder score on a composite scoring rule results in the $ξ$-Hölder divergence.

Relationship between Hölder Divergence and Functional Density Power Divergence: Intersection and Generalization

TL;DR

This paper investigates the relationship between Hölder divergence and the functional density power divergence (FDPD) within the broader landscape of density-power-based divergences. It establishes that the intersection of Hölder divergence and FDPD collapses to the Jones–Hölder–Héroux–Jones (JHHB) divergence family and introduces the -Hölder divergence as a unifying generalization informed by Hölder’s inequality and composite scoring rules. The authors derive an inequality between composite scoring rules corresponding to different FDPDs under the -Hölder framework and show that imposing the Hölder score structure on a composite scoring rule yields the -Hölder divergence. The work clarifies how Hölder divergence and FDPD relate to known divergences (e.g., , -divergence) as special cases and points to future extensions to negative and non-composite divergences, with potential implications for robust statistical inference.

Abstract

In this study, we discuss the relationship between two families of density-power-based divergences with functional degrees of freedom -- the Hölder divergence and the functional density power divergence (FDPD) -- based on their intersection and generalization. These divergence families include the density power divergence and the -divergence as special cases. First, we prove that the intersection of the Hölder divergence and the FDPD is limited to a general divergence family introduced by Jones et al. (Biometrika, 2001). Subsequently, motivated by the fact that Hölder's inequality is used in the proofs of nonnegativity for both the Hölder divergence and the FDPD, we define a generalized divergence family, referred to as the -Hölder divergence. The nonnegativity of the -Hölder divergence is established through a combination of the inequalities used to prove the nonnegativity of the Hölder divergence and the FDPD. Furthermore, we derive an inequality between the composite scoring rules corresponding to different FDPDs based on the -Hölder divergence. Finally, we prove that imposing the mathematical structure of the Hölder score on a composite scoring rule results in the -Hölder divergence.

Paper Structure

This paper contains 18 sections, 11 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Related Divergences
  4. Hölder divergence
  5. Functional density power divergence
  6. Intersection of Hölder divergence and FDPD
  7. Generalization of Hölder divergence and FDPD
  8. Conclusion and future directions
  9. Acknowledgments.
  10. Disclosure of Interests.
  11. Conditions for FDPD on nonnegative functions
  12. Proof of Theorem \ref{['thm:fdpd_affine']}
  13. $\gamma>0$
  14. $\gamma=0$
  15. Proof of Theorem \ref{['thm:hd_jones_div']}
  16. ...and 3 more sections

Key Result

theorem thmcountertheorem

The FDPD is affine invariant if and only if the function $\varphi$ is given by $(z^\zeta-1)/\zeta$ for $\zeta>0$, or $\log z$. The scale function of the affine transformation is given by $h(\bm{\Sigma},\bm{\mu}) =|\det \bm{\Sigma}|^{-\gamma\zeta}$ for $\gamma>0$ or $h(\bm{\Sigma},\bm{\mu}) =|\det \b

Figures (1)

  • Figure 1: Relationship among the DPD, the $\gamma$-divergence, and their generalized divergence families.

Theorems & Definitions (23)

  • definition thmcounterdefinition: Composite scoring rule and divergence holder-div1
  • definition thmcounterdefinition: Equivalence of composite scoring rules holder-div1
  • definition thmcounterdefinition: Affine invariant divergence holder-div1
  • definition thmcounterdefinition: Hölder score and divergence holder-div1
  • definition thmcounterdefinition: Functional density power score and divergence FDPD
  • remark thmcounterremark
  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • proof
  • theorem thmcountertheorem
  • ...and 13 more