On the Structure of the Least Favorable Prior Distributions

Alex Dytso, H. Vincent Poor, Ronit Bustin, Shlomo Shamai

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

10 Scopus citations


This paper studies optimization of the minimum mean square error (MMSE) in order to characterize the structure of the least favorable prior distributions. In the first part, the paper characterizes the local behavior of the MMSE in terms of the input distribution and finds the directional derivative of the MMSE at the distribution P-{\mathbf{X}} in the direction of the distribution Q-{\mathbf{X}}. In the second part of the paper, the directional derivative together with the theory of convex optimization is used to characterize the structure of least favorable distributions. In particular, under mild regularity conditions, it is shown that the support of the least favorable distributions must necessarily be very small and is contained in a nowhere dense set of Lebesgue measure zero. The results of this paper produce both sufficient and necessary conditions for optimality, do not rely on Gaussian statistics assumptions, and are not sensitive to the dimensionality of random vectors. The results are evaluated for the univariate and multivariate random Gaussian cases, and the Poisson case. Finally, as one of the applications, it is shown how the results can be used to characterize the capacity of Gaussian MIMO channels with an amplitude constraint.

Original languageEnglish (US)
Title of host publication2018 IEEE International Symposium on Information Theory, ISIT 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages5
ISBN (Print)9781538647806
StatePublished - Aug 15 2018
Event2018 IEEE International Symposium on Information Theory, ISIT 2018 - Vail, United States
Duration: Jun 17 2018Jun 22 2018

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8095


Other2018 IEEE International Symposium on Information Theory, ISIT 2018
Country/TerritoryUnited States

All Science Journal Classification (ASJC) codes

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


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