对数凹性与强对数凹性综述。

IF 11 Q1 STATISTICS & PROBABILITY

Statistics Surveys Pub Date : 2014-01-01 Epub Date: 2014-12-09 DOI:10.1214/14-SS107

Adrien Saumard, Jon A Wellner

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引用次数: 237

摘要

我们回顾并表述了离散和连续两种情况下关于对数凹性和强对数凹性的结果。我们展示了如何从Efron(1969)的一个基本单调性结果出发，在卷积条件下保持对数凹性和强对数凹性。我们使用Otto和Menz(2013)最近提出的不对称Brascamp-Lieb不等式提供了Efron定理的新证明。在此过程中，我们回顾了对数凹性与其他数学和统计学领域之间的联系，包括测度的集中、对数-索博列夫不等式、凸几何、MCMC算法、拉普拉斯近似和机器学习。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

Log-Concavity and Strong Log-Concavity: a review.

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Log-Concavity and Strong Log-Concavity: a review.

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

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来源期刊

Statistics Surveys STATISTICS & PROBABILITY-

CiteScore

11.70

自引率

0.00%

发文量

期刊介绍： Statistics Surveys publishes survey articles in theoretical, computational, and applied statistics. The style of articles may range from reviews of recent research to graduate textbook exposition. Articles may be broad or narrow in scope. The essential requirements are a well specified topic and target audience, together with clear exposition. Statistics Surveys is sponsored by the American Statistical Association, the Bernoulli Society, the Institute of Mathematical Statistics, and by the Statistical Society of Canada.