Foundations and Trends® in Communications and Information Theory > Vol 17 > Issue 4

Polynomial Methods in Statistical Inference: Theory and Practice

Yihong Wu, Department of Statistics and Data Science, Yale University, USA, yihong.wu@yale.edu , Pengkun Yang, Department of Electrical Engineering, Princeton University, USA, pengkuny@princeton.edu
 
Suggested Citation
Yihong Wu and Pengkun Yang (2020), "Polynomial Methods in Statistical Inference: Theory and Practice", Foundations and TrendsĀ® in Communications and Information Theory: Vol. 17: No. 4, pp 402-586. http://dx.doi.org/10.1561/0100000095

Publication Date: 12 Oct 2020
© 2020 Yihong Wu and Pengkun Yang
 
Subjects
Information theory and statistics
 

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In this article:
1. Introduction
2. Background
3. Polynomial Approximation Methods
4. Entropy Estimation
5. Estimating the Unseen
6. Mixture Models and Moment Comparison Theorems
7. Learning Gaussian Mixtures
Acknowledgements
References

Abstract

This survey provides an exposition of a suite of techniques based on the theory of polynomials, collectively referred to as polynomial methods, which have recently been applied to address several challenging problems in statistical inference successfully. Topics including polynomial approximation, polynomial interpolation and majorization, moment space and positive polynomials, orthogonal polynomials and Gaussian quadrature are discussed, with their major probabilistic and statistical applications in property estimation on large domains and learning mixture models. These techniques provide useful tools not only for the design of highly practical algorithms with provable optimality, but also for establishing the fundamental limits of the inference problems through the method of moment matching. The effectiveness of the polynomial method is demonstrated in concrete problems such as entropy and support size estimation, distinct elements problem, and learning Gaussian mixture models.

DOI:10.1561/0100000095
ISBN: 978-1-68083-730-8
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Table of contents:
1. Introduction
2. Background
3. Polynomial Approximation Methods
4. Entropy Estimation
5. Estimating the Unseen
6. Mixture Models and Moment Comparison Theorems
7. Learning Gaussian Mixtures
Acknowledgements
References

Polynomial Methods in Statistical Inference: Theory and Practice

The authors of this monograph survey a suite of techniques based on the theory of polynomials, collectively referred to as polynomial methods. These techniques provide useful tools not only for the design of highly practical algorithms with provable optimality, but also for establishing the fundamental limits of inference problems through moment matching. The authors demonstrate the effectiveness of the polynomial method using concrete problems such as entropy and support size estimation, distinct elements problem, and learning Gaussian mixture models.

This monograph provides a comprehensive, yet concise, overview of the theory covering topics such as polynomial approximation, polynomial interpolation and majorization, moment space and positive polynomials, orthogonal polynomials and Gaussian quadrature. The authors proceed to show the applications of the theory in statistical inference.

Polynomial Methods in Statistical Inference provides students, and researchers with an accessible and complete treatment of a subject that has recently been used to solve many challenging problems in statistical inference.

 
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