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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 :)

JustInTime
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    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 $X=A+B, Y=A+C$. – kjetil b halvorsen Oct 29 '15 at 13:15
  • 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? https://en.wikipedia.org/wiki/Dirichlet_distribution – Mike Hunter Oct 29 '15 at 13:23
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    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? – kjetil b halvorsen Oct 29 '15 at 13:27
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    Have a look at http://stat-athens.aueb.gr/~karlis/multivariate%20Poisson%20models.pdf – kjetil b halvorsen Oct 29 '15 at 13:42
  • http://stats.stackexchange.com/questions/108705/deriving-the-bivariate-poisson-distribution – kjetil b halvorsen Oct 29 '15 at 14:19
  • @kjetil: Thanks very much for the proposal. Infact independence is dangerous assumption in our setup because most of our features are dependent. Therefore, I hesitated to take such path. – JustInTime Oct 29 '15 at 22:24
  • @DJohnson & Kjetil: Dirichlet is a good idea, Do you know or have a reference of an application or publication where Dirichelt alongside multinomail is used for inference? – JustInTime Oct 29 '15 at 22:31
  • I think you need to give us more details of your actual application to make us advance! – kjetil b halvorsen Oct 30 '15 at 08:32
  • I think people do fit such multivariate Poisson models, but the dependence would tend to be induced through latent variable models. For example, we might set $Y_i \sim \text{Poisson}(e^{\lambda_i})$ and then model $\lambda$ as multivariate Gaussian. This is very common in spatial count models, for example. – guy Jul 25 '20 at 06:28
  • 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. – John Zach Jan 26 '18 at 21:51

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