Concentration of the information in data with log-concave distributions

By Sergey G. Bobkov, Mokshay M. Madiman. 2010

87 citations recorded by the source corpus.

on which the distribution of Xis supported. In this case, eh(X) is essentially the number of bits neededto represent X by a coding scheme that minimizes average code length([Sha48]). In the continuous case (with reference measure dx), one may stillcall eh(X) the information content even though the coding interpretation nolonger holds. In statistics, one may think of the information content as thelog likelihood function.The average value of the information content of Xis known more com-monly as the entropy. Indeed, the entropy of Xis deﬁned byh(X) = −Zf(x)logf(x)dx= −Elogf(X).

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/201b2876c25ddf1df7ae85f99f3caaa298958638

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

A Characterization of Guesswork on Swiftly Tilting Curves (2018, background): “[40] S. Bobkov and M. M. Madiman, “An equipartition property for high-dimensional log-concave distributions.” in Allerton Conference, 2012, pp. 482–488.”
Further Analysis of Outlier Detection with Deep Generative Models (2020, unknown): “a natural image distribution) which can be defined using the notion of weak typicality [13], we are only aware of concentration results for log-concave distributions [14], or for stationary ergodic processes [13].”
Logarithmic bounds for isoperimetry and slices of convex sets (2023, unknown): “Hence, from (40) we obtain an alternative proof of the varentropy inequality of Bobkov and Madiman [6] and Nguyen [38], which is the inequality: V arμ(ψ) ≤ ‖∇ψ‖(2)H−1(μ) = n.”
Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures (2011, background): “Some refinements of these ideas, related to an asymptotic equipartition property for log-concave measures, are described in [13].”
Log-Concavity and Strong Log-Concavity: a review (2014, background): “In Bobkov and Madiman (2011) the concavity of the function I(p) = f(F−1(p)) defined in point (c) of Proposition 2.22, plays a role in the proof of concentration and moment inequalities for the following information quantity: − log f(X) where X is a random vector with log-concave density f .”
Mixed integrals and related inequalities (2012, background): “A known result ([3], see also [2]) states that if f : [0,∞) → [0,∞) is logconcave, then the function”