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

3 Scopus citations


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
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


Conference2021 IEEE Information Theory Workshop, ITW 2021
CityVirtual, Online

Bibliographical note

Funding Information:
Ma Jing1, Liu Mengran1, Shao Xiaoying1, Dai Zheng1, Liang Xiaofeng1, Wang Tong2, Zhang Wenping2, Luo Huiming1 1Department of Education and Training, Chinese Center for Disease Control and Prevention, Beijing 102206, China;2School of Public Health, Shanxi Medical University, Taiyuan 030012, China Corresponding author: Luo Huiming, Email: luohm@chinacdc.cn 【Abstract】 Under the background that the national health has become the national priority development strategy, it is essential to speed up public health ability among talents. Based on the consulting project of the Chinese Academy of Engineering, "Research on the training strategy of medical and health personnel in China", this paper analyzes the current situation and existing problems of public health personnel training in disease control and prevention institutions. Based on three stages of public health education, this paper puts forward that the public health personnel training should first solve the problem of public health personnel team construction and create programs on college education-post graduate education-continuing education. Through the personnel training system, different training modes are designed for other groups of people to improve the ability of public health personnel in an all-around way. 【Key words】 Disease control and prevention; Public health; Personnel training Fund program: Major Consulting and Research Projects of Chinese Academy of Engineering (2016-ZD-11-04)

Publisher Copyright:
© 2021 IEEE.


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