A new Dirichlet-multinomial mixture model for count data

The Dirichlet-multinomial is one of the most known compound distributions for multivariate count data. Let X|p ∼ multinomial(n,p) and P ∼ Dirichlet(α), then the marginal distribution of X is the Dirichlet-multinomial distribution. Because of the severe covariance structure imposed by the Dirichlet prior, covariance among distinct elements of 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 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.