# Compound Poisson Approximation via Information Functionals

@article{Barbour2010CompoundPA, title={Compound Poisson Approximation via Information Functionals}, author={Andrew D. Barbour and Oliver Johnson and Ioannis Kontoyiannis and Mokshay M. Madiman}, journal={ArXiv}, year={2010}, volume={abs/1004.3692} }

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 derived for the distance between the distribution of a sum of independent integer-valued random variables and an appropriately chosen compound Poisson law. In the case where all summands have the same conditional distribution given that they are non-zero, a bound on the relative entropy distance…

## 38 Citations

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

- Computer Science, Mathematics2012 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.

### A criterion for the compound poisson distribution to be maximum entropy

- Mathematics2009 IEEE International Symposium on Information Theory
- 2009

It is shown that the compound Poisson does indeed have a natural maximum entropy characterization when the distributions under consideration are log-concave, which complements the recent development by the same authors of an information-theoretic foundation for compoundPoisson approximation inequalities and limit theorems.

### Information in probability: Another information-theoretic proof of a finite de Finetti theorem

- Computer ScienceArXiv
- 2022

An upper bound on the relative entropy is derived between the distribution of the distribution in a sequence of exchangeable random variables, and an appropriate mixture over product distributions, using de Finetti’s classical representation theorem as a corollary.

### Entropy bounds for discrete random variables via coupling

- Mathematics, Computer Science2013 IEEE International Symposium on Information Theory
- 2013

New bounds on the difference between the entropies of two discrete random variables in terms of the local and total variation distances between their probability mass functions are provided.

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

- MathematicsDiscret. Appl. Math.
- 2013

### Mutual Information, Relative Entropy, and Estimation in the Poisson Channel

- Computer ScienceIEEE Transactions on Information Theory
- 2012

Relative entropy quantifies the excess estimation loss due to mismatch in this setting, parallel to those recently found for the Gaussian channel: the I-MMSE relationship of Guo, the relative entropy and mismatched estimation relationship of Verdú, and the relationship between causal and noncasual mismatch estimation of Weissman.

### Improved lower bounds on the total variation distance and relative entropy for the Poisson approximation

- Mathematics, Computer Science2013 Information Theory and Applications Workshop (ITA)
- 2013

New lower bounds on the total variation distance between the distribution of a sum of independent Bernoulli random variables and the Poisson random variable (with the same mean) are derived via the…

### Compound Poisson Approximation to Convolutions of Compound Negative Binomial Variables

- Mathematics
- 2014

In this paper, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered. First, we obtain an error…

### Accurate Inference for the Mean of the Poisson-Exponential Distribution

- Mathematics
- 2020

Although the random sum distribution has been well-studied in probability theory, inference for the mean of such distribution is very limited in the literature. In this paper, two approaches are…

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