Poisson binomial distribution

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.

Poisson binomial
Parameters — success probabilities for each of the n trials
Support k ∈ {0, …, n}
pmf
CDF
Mean
Variance
Skewness
Ex. kurtosis
MGF
CF

In other words, it is the probability distribution of the number of successes in a sequence of n independent yes/no experiments with success probabilities . The ordinary binomial distribution is a special case of the Poisson binomial distribution, when all success probabilities are the same, that is .

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:

For fixed values of the mean () 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 as n tends to infinity.

Probability mass function

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

where is the set of all subsets of k integers that can be selected from {1,2,3,...,n}. For example, if n = 3, then . is the complement of , i.e. .

will contain elements, the sum over which is infeasible to compute in practice unless the number of trials n is small (e.g. if n = 30, contains over 1020 elements). However, there are other, more efficient ways to calculate .

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]

where

The recursive formula is not numerically stable, and should be avoided if is greater than approximately 20. Another possibility is using the discrete Fourier transform.[4]

where and .

Still other methods are described in [5] .

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 .[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 , if all . 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:

where we took . This is similar to the tail bounds of a binomial distribution.

See also

References

  1. Wang, Y. H. (1993). "On the number of successes in independent trials" (PDF). Statistica Sinica. 3 (2): 295–312.
  2. Shah, B. K. (1994). "On the distribution of the sum of independent integer valued random variables". American Statistician. 27 (3): 123–124. JSTOR 2683639.
  3. Chen, X. H.; A. P. Dempster; J. S. Liu (1994). "Weighted finite population sampling to maximize entropy" (PDF). Biometrika. 81 (3): 457. doi:10.1093/biomet/81.3.457.
  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. Bibcode:2010ITAES..46..803F. doi:10.1109/TAES.2010.5461658.
  5. Chen, S. X.; J. S. Liu (1997). "Statistical Applications of the Poisson-Binomial and conditional Bernoulli distributions". Statistica Sinica. 7: 875–892.
  6. Harremoës, P. (2001). "Binomial and Poisson distributions as maximum entropy distributions" (PDF). IEEE Transactions on Information Theory. 47 (5): 2039–2041. doi:10.1109/18.930936.
  7. Shepp, Lawrence; Olkin, Ingram (1981). "Entropy of the sum of independent Bernoulli random variables and of the multinomial distribution". In Gani, J.; Rohatgi, V.K. (eds.). Contributions to probability: A collection of papers dedicated to Eugene Lukacs. New York: Academic Press. pp. 201–206. ISBN 0-12-274460-8. MR 0618689.
  8. Hillion, Erwan; Johnson, Oliver (2015-03-05). "A proof of the Shepp-Olkin entropy concavity conjecture". Bernoulli. 23: 3638–3649. arXiv:1503.01570. 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.
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