Weighted Gaussian entropy and determinant inequalities

Abstract

We produce a series of results extending information-theoretical inequalities (discussed by Dembo–Cover–Thomas 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 probability, the weighted (or context-dependent) entropy takes into account a ‘value’ of an outcome determined by a given weight function $${\varphi }$$ . An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definite $$d\times d$$ matrix and its square blocks (sub-matrices) of different sizes. When $${\varphi }\equiv 1$$ , the weighted inequality becomes a ‘standard’ SHI; in general, the weighted version requires some assumptions upon $${\varphi }$$ . The SHI and its weighted version generalize a widely known ‘usual’ Hadamard inequality $${\mathrm{det}}\,{{\mathbf {C}}}\le \prod \nolimits _{j=1}^dC_{jj}$$ .