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math.ST updates on arXiv.org

What is Learnable in Valiant's Theory of the Learnable? 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Log-concavity and strong log-concavity: a review
Adrien Saumard, Jon A. Wellner · 2014-04-24 · via math.ST updates on arXiv.org

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 $\mathbb{R}$ 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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