Log-concavity and strong log-concavity: a review

2014/04/23 by Saumard, Adrien, Wellner, Jon A. · 13 citations
#60E15 #62E10 #62H05 #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1404.5886

Abstract

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.

Cited by

• An example of a non-log-concave distribution where the difference has a log-concave density
• Shape-constrained density estimation with Wasserstein projection
• Time-Averaged Drift Approximations are Inconsistent for Inference in Drift Diffusion Models
• Log-concave functions and transformations thereof
• An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows
• Average Winding Number for Determinantal Curves associated with 2-Matrix Models in the Class AIII
• Wasserstein Convergence of Critically Damped Langevin Diffusions
• Polynomial Mixing Times of Simulated Tempering for Mixture Targets by Conductance Decomposition
• A note on the Littlewood-Offord problem for discrete log-concave distributions
• An Improved Bipartition Cover Bound for the Multispecies Coalescent Model
• Expectation-propagation for Bayesian empirical likelihood inference
• Zeroth-order parallel sampling
• A transport approach to the cutoff phenomenon

Related

• Concentration inequalities for log-concave sequences
• Strong Log-Concavity Does Not Imply Log-Submodularity
• New characterizations of log-concavity
• Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension
• On the log-Sobolev constant of log-concave measures