Weighted Gaussian entropy and determinant inequalities
We produce a series of results extending information-theoretical inequalities (dis-
cussed by Dembo and CoverThomas in 1988-1991) to a weighted version of entropy.
Most of the resulting inequalities involve the Gaussian weighted entropy; they imply
a number of new relations for determinants of positive-definite matrices. Unlike the
Shannon entropy where the contribution of an outcome depends only upon its prob-
ability, the weighted (or context-dependent) entropy takes into account a `value' of
an outcome determined by a given weight function '. An example of a new result is a
weighted version of the strong Hadamard inequality (SHI) between the determinants
of a positive-definitenite d by d matrix and its square blocks (sub-matrices) of different
size. When the weight equals 1, the weighted inequality becomes a `standard' SHI; in general,
the weighted version requires some assumptions upon '. The SHI and its weighted
version generalize a widely known `usual' Hadamard inequality.