Radon-Nikodym derivatives of a class of weak distributions on Hilbert spaces

Abstract

The problem of derivatives of weak distributions is studied in the context of likelihood ratios of signals in noise, the ‘independent’ case. We show that the derivative is defined in that case and obtain a formula for it. The main result is in Section 2; the necessary introductory material is in Section 1. The application to the linear case is given in Section 3, and in Section 4, a non-linear example, in which we show for the first time that the correction term in the white noise version of the Girsanov formula is a random variable whose expected value is the mean square estimation error.