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While there are a lot of long tail alternatives to poisson-gamma (negative binomial), for example

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(Source)

I haven't found any work on replacing the dirichlet distribution with a more long tailed alternative (if exists) in the dirichlet-multinomial

The questions are:

  • am I missing some?
  • why is this the case, what is the real mathematical/statistical/numerical difficulty of going beyond the dirichlet?
Sven Hohenstein
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Stefano Vespucci
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  • maybe consider [Aitchison/softmax transformations](https://en.wikipedia.org/wiki/Aitchison_geometry) in conjunction with heavy tailed variants of multivariate normal (multivariate t, multivariate skew-t ...) ? – Ben Bolker Nov 22 '18 at 02:40
  • What do you mean by long trails in Dirichlet? It has fixed support, so it would never have long trails in the usual sense. If you set parameters of Beta or Dirichlet to get high or low values, they result in distributions getting narrowly focused on particular values, isn't this what you're talking about? – Tim Nov 22 '18 at 05:33
  • @Tim, Intuitively I mean that may discrete probability distributions (multinomial, dirichlet-multinomial) are not robust to real-world outliers (from experience and from literature). To my knowledge there no multinomial framework with such propertiy. – Stefano Vespucci Nov 23 '18 at 03:14
  • @BenBolker, thanks. However to my understanding transforming count data to proportions and using continuous distributions will model poorly the 0's given to multinomial process (e.g., many colors of balls but few balls drawn). I am curious why are you talking about multivariate normals and not normals? Is because you want to incorporate the inverse correlation? – Stefano Vespucci Nov 23 '18 at 03:14
  • the idea would be to set up a multivariate normal latent variable in the transformed space (i.e., an N-1-dimensional MVN); then use this variable (transformed to N-dimensional compositional space) as the shape parameters of a Dirichlet distribution. MVN-Dirichlet is automatically overdispersed; MVt-Dirichlet would be heavy-tailed... – Ben Bolker Nov 23 '18 at 03:28
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    See https://stats.stackexchange.com/questions/220543/generate-a-random-set-of-numbers-with-fixed-sum-and-desired-means-and-variances/300500#300500 as an example of what @BenBolker is talking about – Tim Nov 23 '18 at 07:33

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