Saumard Adrien, Wellner Jon A
Departamento de Estadística, CIMFAV, Universidad de Valparaíso, Chile.
Department of Statistics, Box 354322, University of Washington, Seattle, WA 98195-4322.
Stat Surv. 2014;8:45-114. doi: 10.1214/14-SS107. Epub 2014 Dec 9.
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.
我们回顾并阐述了离散和连续情形下关于对数凹性和强对数凹性的结果。我们展示了卷积下实数域上对数凹性和强对数凹性的保持如何从埃弗龙（1969 年）的一个基本单调性结果推导得出。我们利用奥托和门茨（2013 年）最近提出的非对称布拉斯坎普 - 利布不等式给出了埃弗龙定理的一个新证明。在此过程中，我们回顾了对数凹性与数学和统计学其他领域之间的联系，包括测度集中、对数 - 索伯列夫不等式、凸几何、马尔可夫链蒙特卡罗算法、拉普拉斯近似以及机器学习。
