A Characterization of probability densities with expected log likelihood

In the class of all densities with constraints E b_i(X) = c,(i = 1,....,q), it is well known that the entropy is maximized by the density belonging to the exponential family specified by b_i(x), i = 1,....,q. Our aim in this note is to generalize theresult by using expected log likelihood.     

Identities and inequalities for the expected values of random matrices and loewner's theory

We consider the identity associated with the expected values of the matrix-valued quadratic forms of two random matrices. As a special case the identity yields the matrix-valued inequality of the Schwarz type. The most important application is for Loewner's theory in the context of the operator-monotone function, and the inequalities based on the concavityconvexity of the matrix-valued functions are derived.