Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures

@article{Johnson2009LogconcavityUA,
  title={Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures},
  author={Oliver Johnson and Ioannis Kontoyiannis and Mokshay M. Madiman},
  journal={ArXiv},
  year={2009},
  volume={abs/0912.0581}
}

Geometric and functional inequalities for log-concave probability sequences

We investigate various geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A

The Entropy Per Coordinate of a Random Vector is Highly Constrained Under Convexity Conditions

New constraints on entropy per coordinate are given for so-called convex or hyperbolic probability measures on Euclidean spaces, which generalize the results under the log-concavity assumption, expose the extremal role of multivariate Pareto-type distributions, and give some applications.

On the entropy of sums of Bernoulli random variables via the Chen-Stein method

  • I. Sason
  • Computer Science, Mathematics
    2012 IEEE Information Theory Workshop
  • 2012
Upper bounds on the error that follows from an approximation of this entropy by the entropy of a Poisson random variable with the same mean are derived and combines elements of information theory with the Chen-Stein method for Poisson approximation.

Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $\mathbb {L}^{1}$

It is well known that some important Markov semi-groups have a “regularization effect” – as for example th hypercontractivity property of the noise operator on the Boolean hypercube or the

An Information-Theoretic Perspective of the Poisson Approximation via the Chen-Stein Method

  • I. Sason
  • Computer Science, Mathematics
    ArXiv
  • 2012
The analysis in this work combines elements of information theory with the Chen-Stein method for the Poisson approximation to derive new lower bounds on the total variation distance and relative entropy between the distribution of the sum of independent Bernoulli random variables and the Poison distribution.

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

A new proof of Efron's theorem is provided using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013) and along the way connections between log-concavity and other areas of mathematics and statistics are reviewed.

Combinatorial Entropy Power Inequalities: A Preliminary Study of the Stam Region

It is shown that the class of fractionally superadditive set functions provides an outer bound to the Stam region, resolving a conjecture of Barron and Madiman.

Reversal of Rényi Entropy Inequalities Under Log-Concavity

A discrete analog of the Rényi entropy comparison due to Bobkov and Madiman is established, and the entropic Rogers-Shephard inequality studied by Madiman and Kontoyannis is investigated.

An Information-Theoretic Perspective of the Poisson Approximation via the Chen-Stein Method Igal

The analysis in this work combines elements of information theory with the Chen-Stein method for the Poisson approximation to derive new lower bounds on the total variation distance and relative entropy between the distribution of the sum of independent Bernoull i random variables and thePoisson distribution.

Quantitative limit theorems via relative log-concavity

. In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures µ and ν such that ν is

References

SHOWING 1-10 OF 63 REFERENCES

On the entropy and log-concavity of compound Poisson measures

It is shown that the natural analog of the Poisson maximum entropy property remains valid if the measures under consideration are log-concave, but that it fails in general.

Log-concavity and the maximum entropy property of the Poisson distribution

The Entropy Per Coordinate of a Random Vector is Highly Constrained Under Convexity Conditions

New constraints on entropy per coordinate are given for so-called convex or hyperbolic probability measures on Euclidean spaces, which generalize the results under the log-concavity assumption, expose the extremal role of multivariate Pareto-type distributions, and give some applications.

On the Entropy of Compound Distributions on Nonnegative Integers

  • Yaming Yu
  • Mathematics
    IEEE Transactions on Information Theory
  • 2009
Two recent results of Johnson (2008) on maximum entropy characterizations of compound Poisson and compound binomial distributions are proved under fewer assumptions and with simpler arguments.

Compound Poisson Approximation via Information Functionals

An information-theoretic development is given for the problem of compound Poisson approximation, which parallels earlier treatments for Gaussian and Poisson approximation. Nonasymptotic bounds are

On the maximum entropy of the sum of two dependent random variables

It is shown that max[h(X+Y)]=h(2X), under the constraints that X and Y have the same fixed marginal density f, if and only if f is log-concave, which should lead to capacity bounds for additive noise channels with feedback.

A strong log-concavity property for measures on Boolean algebras

Negative dependence and the geometry of polynomials

We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers

Entropy Computations via Analytic Depoissonization

It is argued that analytic methods can offer new tools for information theory, especially for studying second-order asymptotics, and there has been a resurgence of interest and a few successful applications of analytic methods to a variety of problems of information theory.

Entropy and set cardinality inequalities for partition‐determined functions

A new notion of partition‐determined functions is introduced, and several basic inequalities are developed for the entropies of such functions of independent random variables, as well as for
...