Cited by 257Statistical Mechanics and EntropyMathematical Inequalities and ApplicationsBayesian Methods and Mixture Models

On the Jensen–Shannon Symmetrization of Distances Relying on Abstract Means

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Nielsen, Frank·2019·Entropy

Abstract

The Jensen–Shannon divergence is a renowned bounded symmetrization of the unbounded Kullback–Leibler divergence which measures the total Kullback–Leibler divergence to the average mixture distribution. However, the Jensen–Shannon divergence between Gaussian distributions is not available in closed form. To bypass this problem, we present a generalization of the Jensen–Shannon (JS) divergence using abstract means which yields closed-form expressions when the mean is chosen according to the parametric family of distributions. More generally, we define the JS-symmetrizations of any distance using parameter mixtures derived from abstract means. In particular, we first show that the geometric mean is well-suited for exponential families, and report two closed-form formula for (i) the geometric Jensen–Shannon divergence between probability densities of the same exponential family; and (ii) the geometric JS-symmetrization of the reverse Kullback–Leibler divergence between probability densities of the same exponential family. As a second illustrating example, we show that the harmonic mean is well-suited for the scale Cauchy distributions, and report a closed-form formula for the harmonic Jensen–Shannon divergence between scale Cauchy distributions. Applications to clustering with respect to these novel Jensen–Shannon divergences are touched upon.

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  • The PoE and MoE can be subsumed under the concept of abstract means.

    “The PoE and MoE can be subsumed under the concept of abstract means (Nielsen, 2019).”

    From Generalized Multimodal ELBO (2021) · cites this paper as Nielsen, 2019

    Section 3 (Definitions; page not provided) derives ordinary statistical mixtures from the arithmetic mean and normalized weighted products of densities from the geometric mean, both under its abstract-mean framework. This supports the mathematical connection, but the paper does not identify these constructions as mixture-of-experts or product-of-experts models or establish that those models in general are subsumed by the framework. (LLM, conf=0.78)

    Experimental — accuracy not yet measured
  • The choice of mean function directly influences the properties of a model.

    “This insight has practical implications, because the choice of mean function directly influences the properties of a model (Nielsen, 2019).”

    From Generalized Multimodal ELBO (2021) · cites this paper as Nielsen, 2019

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  1. [56]Tzagkarakis G. Tsakalides P. A statistical approach to texture image retrieval via alpha-stable modeling of wavelet decompositions Proceedings of the 5th International Workshop on Image Analysis for Multimedia Interactive Services, Instituto Superior Técnico Lisboa, Portugal 21–23 April 2004 21 23

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Authors
Nielsen, F.
Affiliations
Sony Computer Science Laboratories (JP)
Corresponding author
Nielsen, Frank
Venue
Entropy
Publisher
MDPI AG
Published
2019-05-11
Volume
21
Issue
5
Pages
485
ISSN
1099-4300 (ISSN-L)
OpenAlex
W3102206498
Cited by
257
References
79
Citations by year
2026: 38 · 2025: 63 · 2024: 40 · 2023: 42 · 2022: 27 · 2021: 30 · 2020: 12 · 2019: 5
FWCI
8.56
Type
journal article
Language
en
Open access
Locations
  • Entropy (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Statistical Mechanics and Entropy
  • Physical Sciences › Mathematics › Applied Mathematics › Mathematical Inequalities and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Methods and Mixture Models
Keywords
Jensen-Shannon divergence, geometric mean, harmonic mean, Kullback-Leibler divergence, exponential family, closed-form expressions, clustering

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