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In previous work the authors defined the k-th order simplicial distancebetween probability distributions which arises naturally from a measure ofdispersion based on the squared volume of random simplices of dimension k. Thistheory is embedded in the wider theory of divergences and distances betweendistributions which includes Kullback-Leibler, Jensen-Shannon, Jeffreys-Bregmandivergence and Bhattacharyya distance. A general construction is given based ondefining a directional derivative of a function $\phi$ from one distribution tothe other whose concavity or strict concavity influences the properties of theresulting divergence. For the normal distribution these divergences can beexpressed as matrix formula for the (multivariate) means and covariances.Optimal experimental design criteria contribute a range of functionals appliedto non-negative, or positive definite, information matrices. Not all candistinguish normal distributions but sufficient conditions are given. The k-thorder simplicial distance is revisited from this aspect and the results areused to test empirically the identity of means and covariances.

Bregman divergences based on optimal design criteria and simplicial measures of dispersion