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Title: A new Dirichlet-multinomial mixture model for count data Authors:  Roberto Ascari - University of Milano-Bicocca (Italy) [presenting]
Sonia Migliorati - University of Milano Bicocca (Italy)
Andrea Ongaro - University of Milano-Bicocca (Italy)
Abstract: The Dirichlet-multinomial is one of the most known compound distributions for multivariate count data. Let $\textbf{X} | \textbf{p} \sim$ multinomial$(n, \textbf{p})$ and $\textbf{P} \sim$ Dirichlet$(\boldsymbol{\alpha})$, then the marginal distribution of $\textbf{X}$ is the Dirichlet-multinomial distribution. Because of the severe covariance structure imposed by the Dirichlet prior, covariance among distinct elements of $\textbf{X}$ assumes only negative values and this could be unrealistic in some particular scenarios. In the literature there exist several other distributions defined on the simplex: a recent proposal is the Extended Flexible Dirichlet (EFD), a generalization of the Dirichlet with a less strict dependence structure. A new distribution for count data, called EFD-multinomial, can be obtained by compounding the multinomial model with an EFD prior on the parameters $\textbf{P}$. Due to the covariance structure of the EFD, it allows for positive dependence for some pairs of count categories. Furthermore, thanks to its finite mixture representation, an EM-based estimation procedure can be derived. Some theoretical properties of the EFD-multinomial distribution are shown, and a preliminary simulation study is performed to evaluate the behavior of the EM-based MLE under several scenarios, including positively correlated counts.