We study information projections with respect to statistical $f$-divergences between any two location-scale families. We consider a multivariate generalization of the location-scale families which includes the elliptical and the spherical subfamilies. By using the action of the multivariate location-scale group, we show how to reduce the calculation of $f$-divergences between any two location-scale densities to canonical settings involving standard densities, and derive thereof fast Monte Carlo estimators of $f$-divergences with good properties. Finally, we prove that the minimum $f$-divergence between a prescribed density of a location-scale family and another location-scale family is independent of the prescribed location-scale parameter. We interpret geometrically this property.

翻译：我们研究任一两地级家庭之间以美元计价的统计信息预测,我们考虑对包括椭圆形和球形次家庭在内的地点级家庭进行多种变式的概括化,通过多变地点级小组的行动,我们展示如何减少任何两个地点级的密度之间的美元差异,以计算标准密度的罐体环境的美元差异,并由此得出具有良好属性的蒙特卡洛快速估计值。最后,我们证明,一个地点级家庭和另一个地点级家庭规定密度之间的最低美元差异,独立于规定的地点级参数。我们从几何角度解释这一属性。