Unifying the Brascamp-Lieb Inequality and the Entropy Power Inequality

The entropy power inequality (EPI) and the Brascamp-Lieb (BLI) are fundamental inequalities concerning differential entropies of linear transformations random vectors. EPI provides lower bounds for vectors with independent components. BLI, on other hand, upper a vector in terms some its transformations. In this paper, we define family functionals, which show subadditive. We then establish that Gaussians extremal these functionals by adapting proof technique from Geng Nair (2014). As consequence, obtain new generalizes both BLI EPI. By considering variety independence relations among components appearing also families lie between BLI.