Poisson binomial distribution

Short description: Probability distribution

Poisson binomialParameters|  [math]\displaystyle{ \mathbf{p}\in [0,1]^n }[/math] — success probabilities for each of the n trials

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Support|  k ∈ { 0, …, n }
pmf|  [math]\displaystyle{ \sum\limits_{A\in F_k} \prod\limits_{i\in A} p_i \prod\limits_{j\in A^c} (1-p_j) }[/math]
CDF|  [math]\displaystyle{ \sum\limits_{l=0}^k \sum\limits_{A\in F_l} \prod\limits_{i\in A} p_i \prod\limits_{j\in A^c} (1-p_j) }[/math]
Mean|  [math]\displaystyle{ \sum\limits_{i=1}^n p_i }[/math]
Variance|  [math]\displaystyle{ \sigma^2 =\sum\limits_{i=1}^n (1 - p_i)p_i }[/math]
Skewness|  [math]\displaystyle{ \frac{1}{\sigma^3}\sum\limits_{i=1}^n ( 1-2p_i ) ( 1-p_i ) p_i }[/math]
Kurtosis|  [math]\displaystyle{ \frac{1}{\sigma^4}\sum\limits_{i=1}^n ( 1 - 6(1 - p_i)p_i )( 1 - p_i )p_i }[/math]
MGF|  [math]\displaystyle{ \prod\limits_{j=1}^n (1-p_j+p_j e^t) }[/math]
CF|  [math]\displaystyle{ \prod\limits_{j=1}^n (1-p_j+p_j e^{it}) }[/math]
PGF|  [math]\displaystyle{ \prod\limits_{j=1}^n (1-p_j+p_j z) }[/math]

In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The concept is named after Siméon Denis Poisson.

In other words, it is the probability distribution of the number of successes in a collection of n independent yes/no experiments with success probabilities [math]\displaystyle{ p_1, p_2, \dots , p_n }[/math]. The ordinary binomial distribution is a special case of the Poisson binomial distribution, when all success probabilities are the same, that is [math]\displaystyle{ p_1 = p_2 = \cdots = p_n }[/math].

## Contents

* 1 Definitions
* 1.1 Probability Mass Function
* 1.2 Cumulative distribution function
* 2 Properties
* 2.1 Mean and Variance
* 2.2 Entropy
* 2.3 Chernoff bound
* 3 Computational methods
* 4 See also
* 5 References

## Definitions

### Probability Mass Function

The probability of having k successful trials out of a total of n can be written as the sum [1]

[math]\displaystyle{ \Pr(K=k) = \sum\limits_{A\in F_k} \prod\limits_{i\in A} p_i \prod\limits_{j\in A^c} (1-p_j) }[/math]

where [math]\displaystyle{ F_k }[/math] is the set of all subsets of k integers that can be selected from [math]\displaystyle{ \\{1,2,3,...,n\\} }[/math]. For example, if n = 3, then [math]\displaystyle{ F_2=\left\\{ \\{1,2\\},\\{1,3\\},\\{2,3\\} \right\\} }[/math]. [math]\displaystyle{ A^c }[/math] is the complement of [math]\displaystyle{ A }[/math], i.e. [math]\displaystyle{ A^c =\\{1,2,3,\dots,n\\}\setminus A }[/math].

[math]\displaystyle{ F_k }[/math] will contain [math]\displaystyle{ n!/((n-k)!k!) }[/math] elements, the sum over which is infeasible to compute in practice unless the number of trials n is small (e.g. if n = 30, [math]\displaystyle{ F_{15} }[/math] contains over 1020 elements). However, there are other, more efficient ways to calculate [math]\displaystyle{ \Pr(K=k) }[/math].

As long as none of the success probabilities are equal to one, one can calculate the probability of k successes using the recursive formula [2] [3]

[math]\displaystyle{ \Pr (K=k)= \begin{cases} \prod\limits_{i=1}^n (1-p_i) & k=0 \\\ \frac{1}{k} \sum\limits_{i=1}^k (-1)^{i-1}\Pr (K=k-i)T(i) & k\gt 0 \\\ \end{cases} }[/math]

where

[math]\displaystyle{ T(i)=\sum\limits_{j=1}^n \left( \frac{p_j}{1-p_j} \right)^i. }[/math]

The recursive formula is not numerically stable, and should be avoided if [math]\displaystyle{ n }[/math] is greater than approximately 20. An alternative is to use a divide-and-conquer algorithm: if we assume [math]\displaystyle{ n = 2^b }[/math] is a power of two, denoting by [math]\displaystyle{ f(p_{i:j}) }[/math] the Poisson binomial of [math]\displaystyle{ p_i, \dots, p_j }[/math] and [math]\displaystyle{ * }[/math] the convolution operator, we have [math]\displaystyle{ f(p_{1:2^b}) = f(p_{1:2^{b-1}})*f(p_{2^{b-1}+1:2^b}) }[/math].

Another possibility is using the discrete Fourier transform.[4]

[math]\displaystyle{ \Pr (K=k)=\frac{1}{n+1} \sum\limits_{l=0}^n C^{-lk} \prod\limits_{m=1}^n \left( 1+(C^l-1) p_m \right) }[/math]

where [math]\displaystyle{ C=\exp \left( \frac{2i\pi }{n+1} \right) }[/math] and [math]\displaystyle{ i=\sqrt{-1} }[/math].

Still other methods are described in "Statistical Applications of the Poisson-Binomial and conditional Bernoulli distributions" by Chen and Liu.[5]

### Cumulative distribution function

The cumulative distribution function (CDF) can be expressed as:

[math]\displaystyle{ \Pr(K \leq k) = \sum^{k}_{l=0} \sum\limits_{A\in F_l} \prod\limits_{i\in A} p_i \prod\limits_{j\in A^c} (1-p_j) }[/math] ,

where [math]\displaystyle{ F_l }[/math] is the set of all subsets of size [math]\displaystyle{ l }[/math] that can be selected from [math]\displaystyle{ \\{1,2,3,...,n\\} }[/math].

## Properties

### Mean and Variance

Since a Poisson binomial distributed variable is a sum of n independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the n Bernoulli distributions:

[math]\displaystyle{ \mu = \sum\limits_{i=1}^n p_i }[/math]

[math]\displaystyle{ \sigma^2 =\sum\limits_{i=1}^n (1-p_i) p_i }[/math]

For fixed values of the mean ([math]\displaystyle{ \mu }[/math]) and size (n), the variance is maximal when all success probabilities are equal and we have a binomial distribution. When the mean is fixed, the variance is bounded from above by the variance of the Poisson distribution with the same mean which is attained asymptotically[citation needed] as n tends to infinity.

### Entropy

There is no simple formula for the entropy of a Poisson binomial distribution, but the entropy is bounded above by the entropy of a binomial distribution with the same number parameter and the same mean. Therefore, the entropy is also bounded above by the entropy of a Poisson distribution with the same mean.[6]

The Shepp–Olkin concavity conjecture, due to Lawrence Shepp and Ingram Olkin in 1981, states that the entropy of a Poisson binomial distribution is a concave function of the success probabilities [math]\displaystyle{ p_1, p_2, \dots , p_n }[/math].[7] This conjecture was proved by Erwan Hillion and Oliver Johnson in 2015.[8] The Shepp-Olkin monotonicity conjecture, also from the same 1981 paper, is that the entropy is monotone increasing in [math]\displaystyle{ p_i }[/math], if all [math]\displaystyle{ p_i \leq 1/2 }[/math]. This conjecture was also proved by Hillion and Johnson, in 2019 [9]

### Chernoff bound

The probability that a Poisson binomial distribution gets large, can be bounded using its moment generating function as follows (valid when [math]\displaystyle{ s \geq \mu }[/math] and for any [math]\displaystyle{ t\gt 0 }[/math]):

[math]\displaystyle{ \begin{align} \Pr[S\ge s] &\le \exp(-st)\operatorname E\left[\exp\left[t \sum_i X_i\right]\right] \\\&= \exp(-st)\prod_i (1-p_i+e^t p_i) \\\&= \exp\left(-st + \sum_i \log\left( p_i (e^t - 1) + 1\right)\right) \\\&\le \exp\left(-st + \sum_i \log\left( \exp(p_i(e^t-1))\right)\right) \\\&= \exp\left(-st + \sum_i p_i(e^t-1)\right) \\\&= \exp\left(s-\mu-s\log\frac{s}{\mu}\right), \end{align} }[/math]

where we took [math]\displaystyle{ t=\log\left(s/\mu\right) }[/math]. This is similar to the tail bounds of a binomial distribution.

## Computational methods

The reference [10] discusses techniques of evaluating the probability mass function of the Poisson binomial distribution. The following software implementations are based on it:

* An R package poibin was provided along with the paper,[10] which is available for the computing of the cdf, pmf, quantile function, and random number generation of the Poisson binomial distribution. For computing the PMF, a DFT algorithm or a recursive algorithm can be specified to compute the exact PMF, and approximation methods using the normal and Poisson distribution can also be specified.
* poibin \- Python implementation \- can compute the PMF and CDF, uses the DFT method described in the paper for doing so.

## See also

* Le Cam's theorem

## References

1. ↑ Wang, Y. H. (1993). "On the number of successes in independent trials". Statistica Sinica 3 (2): 295–312. http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf.
2. ↑ Shah, B. K. (1994). "On the distribution of the sum of independent integer valued random variables". American Statistician 27 (3): 123–124.
3. ↑ Chen, X. H.; A. P. Dempster; J. S. Liu (1994). "Weighted finite population sampling to maximize entropy". Biometrika 81 (3): 457. doi:10.1093/biomet/81.3.457. http://www.people.fas.harvard.edu/~junliu/TechRept/94folder/cdl94.pdf.
4. ↑ Fernandez, M.; S. Williams (2010). "Closed-Form Expression for the Poisson-Binomial Probability Density Function". IEEE Transactions on Aerospace and Electronic Systems 46 (2): 803–817. doi:10.1109/TAES.2010.5461658. Bibcode: 2010ITAES..46..803F.
5. ↑ Chen, S. X.; J. S. Liu (1997). "Statistical Applications of the Poisson-Binomial and conditional Bernoulli distributions". Statistica Sinica 7: 875–892. http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=7&num=4&art=4.
6. ↑ Harremoës, P. (2001). "Binomial and Poisson distributions as maximum entropy distributions". IEEE Transactions on Information Theory 47 (5): 2039–2041. doi:10.1109/18.930936. http://www.harremoes.dk/Peter/poisson.pdf.
7. ↑ Gani, J.; Rohatgi, V.K., eds (1981). "Entropy of the sum of independent Bernoulli random variables and of the multinomial distribution". Contributions to probability: A collection of papers dedicated to Eugene Lukacs. New York: Academic Press. pp. 201–206. ISBN 0-12-274460-8. https://books.google.com/books?id=9qziBQAAQBAJ.
8. ↑ Hillion, Erwan; Johnson, Oliver (2015-03-05). "A proof of the Shepp-Olkin entropy concavity conjecture". Bernoulli 23 (4B): 3638–3649. doi:10.3150/16-BEJ860.
9. ↑ Hillion, Erwan; Johnson, Oliver (2019-11-09). "A proof of the Shepp-Olkin entropy monotonicity conjecture". Electronic Journal of Probability 24 (126): 1–14. doi:10.1214/19-EJP380.
10. ↑ 10.0 10.1 Hong, Yili (March 2013). "On computing the distribution function for the Poisson binomial distribution". Computational Statistics & Data Analysis 59: 41–51. doi:10.1016/j.csda.2012.10.006.

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