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On the Jensen–Shannon Symmetrization of Distances Relying on Abstract Means
Open accessAbstract
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 (Nielsen, 2019).”
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“This insight has practical implications, because the choice of mean function directly influences the properties of a model (Nielsen, 2019).”
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Metadata
- Authors
- Nielsen, F.
- Affiliations
- Sony Computer Science Laboratories (JP)
- Corresponding author
- Nielsen, Frank
- ORCID
- Nielsen, Frank
- Venue
- Entropy
- Publisher
- MDPI AG
- Published
- 2019-05-11
- Volume
- 21
- Issue
- 5
- Pages
- 485
- ISSN
- 1099-4300 (ISSN-L)
- PMC
- PMC7514974
- 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
- gold
- dx.doi.org
- Licence
- 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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