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 - Publisher Copyright:
© 2020 IEEE.
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 -