The entropic centers of multivariate normal distributions

In this paper, we seek for a single best representative of a set of statistical multivariate normal distributions. To define the “best” center, we consider either minimizing the average or the maximum relative entropy of the center to the given set of normal distributions. Since the relative entropy is an asymmetric divergence, this yields the notion of leftand right-sided, and symmetrized entropic centroids and circumcenters along with their respective information radii. We show how to instantiate and implement for this special case of multivariate normals very recent work that tackled the broader case of finding centers of point sets with respect to Bregman divergences. 1 Information-theoretic centers Consider a set of n multivariate normal distributions D = {N(μ1,Σ1), ..., N(μn,Σn)} with μi ∈ R denoting the mean vector and Σi the d × d symmetric positive semi-definite variancecovariance matrix (i.e., xΣix ≥ 0 for all x ∈ R). The probability density function Pr(X = x) = p(x;μ,Σ) of a normal random variable X ∼ N(μ,Σ) is given as: