2019/05/11 by Frank Nielsen · 2 citations
Physics and Astronomy · Mathematics · #Statistical Mechanics and Entropy #Mathematical Inequalities and Applications #Advanced Statistical Methods and Models #Mathematics #Divergence (linguistics) #Kullback–Leibler divergence #Symmetrization #Cauchy distribution #Rényi entropy #Exponential family #Information theory #Pure mathematics #Mathematical analysis #Applied mathematics #Statistics #Principle of maximum entropy
paper · pdf · doi:10.3390/e21050485
openalex publication_date 2019/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.