A General Derivative Identity for the Conditional Expectation with Focus on the Exponential Family

Alex Dytso, Martina Cardone

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Scopus citations

Abstract

Consider a pair of random vectors (X, Y) and the conditional expectation operator bE[X Y=y]. This work studies analytic properties of the conditional expectation by characterizing various derivative identities. The paper consists of two parts. In the first part of the paper, a general derivative identity for the conditional expectation is derived. Specifically, for the Markov chain U ← X ← Y, a compact expression for the Jacobian matrix of bE[U Y=y] is derived. In the second part of the paper, the main identity is specialized to the exponential family. Moreover, via various choices of the random vector U, the new identity is used to recover and generalize several known identities and derive some new ones. As a first example, a connection between the Jacobian of bE[X Y=y] and the conditional variance is established. As a second example, a recursive expression between higher order conditional expectations is found, which is shown to lead to a generalization of the Tweedy's identity. Finally, as a third example, it is shown that the k-th order derivative of the conditional expectation is proportional to the (k+1)-th order conditional cumulant.

Original languageEnglish (US)
Title of host publication2021 IEEE Information Theory Workshop, ITW 2021 - Proceedings
PublisherInstitute of Electrical and Electronics Engineers Inc.
ISBN (Electronic)9781665403122
DOIs
StatePublished - 2021
Event2021 IEEE Information Theory Workshop, ITW 2021 - Virtual, Online, Japan
Duration: Oct 17 2021Oct 21 2021

Publication series

Name2021 IEEE Information Theory Workshop, ITW 2021 - Proceedings

Conference

Conference2021 IEEE Information Theory Workshop, ITW 2021
Country/TerritoryJapan
CityVirtual, Online
Period10/17/2110/21/21

Bibliographical note

Publisher Copyright:
© 2021 IEEE.

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