A General Derivative Identity for the Conditional Mean Estimator in Gaussian Noise and Some Applications

This paper provides a general derivative identity for the conditional mean estimator of an arbitrary vector signal in Gaussian noise with an arbitrary covariance matrix. This new identity is used to recover and generalize many known identities in the literature and derive some new identities. For example, a new identity is discovered, which shows that an arbitrary higher-order conditional moment is completely determined by the first conditional moment.Several applications of the identities are shown. For instance, by using one of the identities, a simple proof of the uniqueness of the conditional mean estimator as a function of the distribution of the signal is shown. Moreover, one of the identities is used to extend the notion of empirical Bayes to higher-order conditional moments. Specifically, based on a random sample of noisy observations, a consistent estimator for a conditional expectation of any order is derived.