%0 Journal Article %1 nielsen2017wmixtures %A Nielsen, Frank %A Nock, Richard %D 2017 %K divergences entropy information readings theory %T On $w$-mixtures: Finite convex combinations of prescribed component distributions %U http://arxiv.org/abs/1708.00568 %X We consider the space of $w$-mixtures that is the set of finite statistical mixtures sharing the same prescribed component distributions. The geometry induced by the Kullback-Leibler (KL) divergence on this family of $w$-mixtures is a dually flat space in information geometry called the mixture family manifold. It follows that the KL divergence between two $w$-mixtures is equivalent to a Bregman Divergence (BD) defined for the negative Shannon entropy generator. Thus the KL divergence between two Gaussian Mixture Models (GMMs) sharing the same components is (theoretically) a Bregman divergence. This KL-BD equivalence implies that we can perform optimal KL-averaging aggregation of $w$-mixtures without information loss. More generally, we prove that the skew Jensen-Shannon divergence between $w$-mixtures is equivalent to a skew Jensen divergence on their parameters. Finally, we state several divergence identity and inequalities relating $w$-mixtures.

@article{nielsen2017wmixtures, abstract = {We consider the space of $w$-mixtures that is the set of finite statistical mixtures sharing the same prescribed component distributions. The geometry induced by the Kullback-Leibler (KL) divergence on this family of $w$-mixtures is a dually flat space in information geometry called the mixture family manifold. It follows that the KL divergence between two $w$-mixtures is equivalent to a Bregman Divergence (BD) defined for the negative Shannon entropy generator. Thus the KL divergence between two Gaussian Mixture Models (GMMs) sharing the same components is (theoretically) a Bregman divergence. This KL-BD equivalence implies that we can perform optimal KL-averaging aggregation of $w$-mixtures without information loss. More generally, we prove that the skew Jensen-Shannon divergence between $w$-mixtures is equivalent to a skew Jensen divergence on their parameters. Finally, we state several divergence identity and inequalities relating $w$-mixtures.}, added-at = {2019-12-11T14:27:59.000+0100}, author = {Nielsen, Frank and Nock, Richard}, biburl = {https://www.bibsonomy.org/bibtex/2f6e4dd0e28b1f800a87d4123e3675b3e/kirk86}, description = {[1708.00568] On $w$-mixtures: Finite convex combinations of prescribed component distributions}, interhash = {a63e35531074c0dd6177d0d5479469d4}, intrahash = {f6e4dd0e28b1f800a87d4123e3675b3e}, keywords = {divergences entropy information readings theory}, note = {cite arxiv:1708.00568Comment: 25 pages}, timestamp = {2019-12-11T14:29:51.000+0100}, title = {On $w$-mixtures: Finite convex combinations of prescribed component distributions}, url = {http://arxiv.org/abs/1708.00568}, year = 2017 }