# Relationship between H"older Divergence and Functional Density Power Divergence: Intersection and Generalization

arXiv: https://arxiv.org/abs/2504.17008 · HTML: https://mlsanity.com/paper/2504.17008/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 16 references that are in the map.
- First posted on arXiv 23 Apr 2025.

## Builds on (most cited first, 3 of 3)

- [On H"older projective divergences](https://mlsanity.com/paper/1701.03916/) (2017)
- [Statistical Inference based on Bridge Divergences](https://mlsanity.com/paper/1706.05745/) (2017)
- [A Unified Representation of Density-Power-Based Divergences Reducible to M-Estimation](https://mlsanity.com/paper/2501.16287/) (2025)

## Abstract (arXiv)

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