  We consider the space of $w$-mixtures which is defined as the set of finite
statistical mixtures sharing the same prescribed component distributions closed
under convex combinations. The information geometry induced by the Bregman
generator set to the Shannon negentropy on this space yields a dually flat
space called the mixture family manifold. We show how the Kullback-Leibler (KL)
divergence can be recovered from the corresponding Bregman divergence for the
negentropy generator: That is, the KL divergence between two $w$-mixtures
amounts to a Bregman Divergence (BD) induced by the Shannon negentropy
generator. Thus the KL divergence between two Gaussian Mixture Models (GMMs)
sharing the same Gaussian components is equivalent to a Bregman divergence.
This KL-BD equivalence on a mixture family manifold implies that we can perform
optimal KL-averaging aggregation of $w$-mixtures without information loss. More
generally, we prove that the statistical skew Jensen-Shannon divergence between
$w$-mixtures is equivalent to a skew Jensen divergence between their
corresponding parameters. Finally, we state several properties, divergence
identities, and inequalities relating to $w$-mixtures.
