Parametric Modelling of Multivariate Count Data Using Probabilistic Graphical Models

Multivariate count data are defined as the number of items of different categories issued from sampling within a population, which individuals are grouped into categories. The analysis of multivariate count data is a recurrent and crucial issue in numerous modelling problems, particularly in the fields of biology and ecology (where the data can represent, for example, children counts associated with multitype branching processes), sociology and econometrics. Denoting by K the number of categories, multivariate count data analysis relies on modelling the joint distribution of the K-dimensional random vector N = (N0, . . . , NK−1) with discrete components. We focus on I) Identifying categories that appear simultaneously, or on the contrary that are mutually exclusive. This is achieved by identifying conditional independence relationships between the K variables; II)Building parsimonious parametric models consistent with these relationships; III) Characterising and testing the effects of covariates on the distribution of N , particularly on the dependencies between its components. To achieve these goals, we propose an approach based on graphical probabilistic models (Koller & Friedman, 2009) to represent the conditional independence relationships in N , and on parametric distributions to ensure model parsimony. Three kinds of graphs are usually considered: either undirected (UG), directed acyclic (DAG), or partially directed acyclic (PDAG). Models and methods for graph identification were proposed in UGs, using frequencies to estimate the probabilities (so-called nonparametric estimation), using mutual information – see Meyer et al. (2008) and references therein. Under a multivariate Gaussian assumption, an approach based on a L1 penalisation (lasso) was proposed by Friedman et al. (2008), with some extension to Poisson distributions with log-linear models for dependencies (Allen & Liu, 2012). Specific models and methods were developed for DAGs. Most methods for graph identification in DAGs are based on exploring the set of possible graphs using some heuristic (e.g. hill climbing) and by scoring the visited graphs (e.g. using BIC), the graph with highest score being eventually selected – see Koller & Friedman (2009) for a review. The case of parametric models for PDAGs has been considered less often in the literature. A family of such models was proposed by Johnson & Hoeting (2011) using conditional Gaussian distributions, but the problem of graph identification was not addressed. Lee & Hastie (2012) addressed the problem of graph identification in graphical models with both continuous and discrete random variables, but in a restrictive setting of UGs with conditional Gaussian and multinomial distributions. Our context of application is characterised by zero-inflated, often right skewed marginal distributions. Thus, Gaussian and Poisson distributions are not a priori appropriate. Moreover, the multivariate histograms typically have many cells, most of which are empty. Consequently, nonparametric estimation is not efficient.