### Abstract

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.

Original language | English (US) |
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Title of host publication | 2020 IEEE International Symposium on Information Theory, ISIT 2020 - Proceedings |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 1183-1188 |

Number of pages | 6 |

ISBN (Electronic) | 9781728164328 |

DOIs | |

State | Published - Jun 2020 |

Event | 2020 IEEE International Symposium on Information Theory, ISIT 2020 - Los Angeles, United States Duration: Jul 21 2020 → Jul 26 2020 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2020-June |

ISSN (Print) | 2157-8095 |

### Conference

Conference | 2020 IEEE International Symposium on Information Theory, ISIT 2020 |
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Country | United States |

City | Los Angeles |

Period | 7/21/20 → 7/26/20 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Information Systems
- Modeling and Simulation
- Applied Mathematics

### Keywords

- Conditional mean estimator
- empirical Bayes
- Gaussian Noise

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## Cite this

*2020 IEEE International Symposium on Information Theory, ISIT 2020 - Proceedings*(pp. 1183-1188). [9174071] (IEEE International Symposium on Information Theory - Proceedings; Vol. 2020-June). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT44484.2020.9174071