TY - GEN

T1 - A General Derivative Identity for the Conditional Mean Estimator in Gaussian Noise and Some Applications

AU - Dytso, Alex

AU - Poor, H. Vincent

AU - Shamai Shitz, Shlomo

N1 - Funding Information:
This work was supported in part by the U. S. National Science Foundation under Grant CCF-1908308 and by the United States-Israel Binational Science Foundation, under Grant BSF-2018710.

PY - 2020/6

Y1 - 2020/6

N2 - 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.

AB - 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.

KW - Conditional mean estimator

KW - Gaussian Noise

KW - empirical Bayes

UR - http://www.scopus.com/inward/record.url?scp=85087474832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85087474832&partnerID=8YFLogxK

U2 - 10.1109/ISIT44484.2020.9174071

DO - 10.1109/ISIT44484.2020.9174071

M3 - Conference contribution

AN - SCOPUS:85087474832

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 1183

EP - 1188

BT - 2020 IEEE International Symposium on Information Theory, ISIT 2020 - Proceedings

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2020 IEEE International Symposium on Information Theory, ISIT 2020

Y2 - 21 July 2020 through 26 July 2020

ER -