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I actually assumed it would be easy to find a multivariate version of the Poisson distribution, but couldn't find any concrete solution (in terms of a well cited publication). It seems that multivariate Poisson models are not commonly used in business applications ("risk management", prediction of rare events). But why is this the case? Am I looking for the wrong thing? Any help is appreciated :)

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  • $\begingroup$ For what application do you want it? There are multiple ways of defining mutivariate Poisson distributions, the most obvious is to let $A,B,C$ be independent (univar) POisson, then define $X=A+B, Y=A+C$. $\endgroup$ – kjetil b halvorsen Oct 29 '15 at 13:15
  • $\begingroup$ The poisson is an appropriate distribution for count or integer data when the variance equals the mean and is useful, to your point, for predicting rare events. As such it sees wide use, e.g., in frequency and severity analyses in actuarial science and risk mgmt. But, by "multivariate" are you referring to a joint probabilistic distribution of multiple, random, poisson process variates? If so, why wouldn't the Dirichlet work? en.wikipedia.org/wiki/Dirichlet_distribution $\endgroup$ – Mike Hunter Oct 29 '15 at 13:23
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    $\begingroup$ How would you use the Dirichlet then? It is a continuous distribution, conjugate prior for the multinomial, for example ... dont see the connection to multivariate counts? $\endgroup$ – kjetil b halvorsen Oct 29 '15 at 13:27
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    $\begingroup$ Have a look at stat-athens.aueb.gr/~karlis/multivariate%20Poisson%20models.pdf $\endgroup$ – kjetil b halvorsen Oct 29 '15 at 13:42
  • $\begingroup$ stats.stackexchange.com/questions/108705/… $\endgroup$ – kjetil b halvorsen Oct 29 '15 at 14:19
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Have a look at https://www.jstor.org/stable/2288540. The authors use a Dirichlet prior on the normalised intensities and a Multinomial likelihood on the counts. The posterior ends up being a Dirichlet distribution. For Bayesian analysts of the Multinomial Dirichlet distribution, have a look at https://people.eecs.berkeley.edu/~stephentu/writeups/dirichlet-conjugate-prior.pdf. Hope that helps.

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