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How to better solve estimation problems

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Veröffentlicht am Freitag, den 19. Februar 2021

Mathematicians from the Universities of Luxembourg and Sorbonne have recently published a paper in the prestigious journal Annals of Statistics. In this paper, they propose a new robust tool for solving estimation problems within the Bayesian paradigm.

The Bayesian posterior distribution was introduced by the British mathematician Thomas Bayes in the 18th century and has since become one of the most popular statistical inference tools worldwide and particularly in the United Kingdom, Australia and the United States. It is widely used in all fields of science and particularly in the life sciences to solve estimation problems. One of the reasons for this popularity is that estimators based on the Bayesian posterior distribution have good estimation properties that can be mathematically grounded and experimentally observed.  

“For the problem of estimating an unknown parameter, Bayes' approach is based on the choice of a prior which is a probabilistic distribution describing the part A of the space in which the parameter is most likely to be located. The prior corresponds to some intuition or guess that a statistician can have on the location of the parameter he or she wants to estimate. Unfortunately, when the parameter is not exactly located in A, which means that the guess is wrong, the Bayes estimator performs poorly, even in the situation where the parameter actually lies close A, which means that our guess is only slightly wrong”, explains Yannick Baraud, Professor in statistics within the Department of Mathematics at the University of Luxembourg and one of the two authors of the paper. 

In this paper, the authors provide a substitute for the very famous Bayesian posterior distribution. This surrogate, named the Rho-Bayes posterior distribution, enjoys the same nice properties as the Bayesian one except from the fact the resulting estimators will still perform well under a slight misspecification of the prior, that is, even when our guess is slightly wrong. The Rho-Bayes posterior distribution results in robust estimators in the Bayesian paradigm.

Article “Robust Bayes-like estimation: Rho-Bayes estimation”, Annals of Statistics, December 2020