%0 Journal Article %1 nielsen2019kullbackleibler %A Nielsen, Frank %D 2019 %K divergences entropy information theory %T On the Kullback-Leibler divergence between location-scale densities %U http://arxiv.org/abs/1904.10428 %X We show that the $f$-divergence between any two densities of potentially different location-scale families can be reduced to the calculation of the $f$-divergence between one standard density with another location-scale density. It follows that the $f$-divergence between two scale densities depends only on the scale ratio. We then report conditions on the standard distribution to get symmetric $f$-divergences. We illustrate this symmetric property with the calculation of the Kullback-Leibler divergence between scale Cauchy distributions. Finally, we show that the minimum $f$-divergence of any query density of a location-scale family to another location-scale family is independent of the query location-scale parameters.

@article{nielsen2019kullbackleibler, abstract = {We show that the $f$-divergence between any two densities of potentially different location-scale families can be reduced to the calculation of the $f$-divergence between one standard density with another location-scale density. It follows that the $f$-divergence between two scale densities depends only on the scale ratio. We then report conditions on the standard distribution to get symmetric $f$-divergences. We illustrate this symmetric property with the calculation of the Kullback-Leibler divergence between scale Cauchy distributions. Finally, we show that the minimum $f$-divergence of any query density of a location-scale family to another location-scale family is independent of the query location-scale parameters.}, added-at = {2019-12-11T14:19:29.000+0100}, author = {Nielsen, Frank}, biburl = {https://www.bibsonomy.org/bibtex/278fb84ae12045d10009cdd8bb323c23b/kirk86}, description = {[1904.10428] On the Kullback-Leibler divergence between location-scale densities}, interhash = {aec989f10002f940e09c77232ccceb3b}, intrahash = {78fb84ae12045d10009cdd8bb323c23b}, keywords = {divergences entropy information theory}, note = {cite arxiv:1904.10428Comment: 11 pages}, timestamp = {2019-12-11T14:19:29.000+0100}, title = {On the Kullback-Leibler divergence between location-scale densities}, url = {http://arxiv.org/abs/1904.10428}, year = 2019 }