BAYESIAN ANALYSIS OF D- DECOMPOSABLE B IDIRECTED GRAPHICAL MODELS FOR DISCRETE DATA Claudia Tarantola 1 and Ioannis Ntzoufras 2 1 Department of Economics and Quantitative methods University of Pavia Via San Felice 5, 27100 Pavia, Italy (e-mail: claudia.tarantola@unipv.it) 2 Department of Statistics Athens University of Economics and Business 76 Patision Street, 10434, Athens, Greece (e-mail: ntzoufras@aueb.gr) ABSTRACT. This paper deals with the Bayesian analysis of d-decomposable graphical models of mar- ginal independence for discrete data. The model is represented by a bidirected graph with missing edges indicating marginal independence between the corresponding variables. A bidirected graphs is named d-decomposable if it is Markov equivalent to at least one DAG. We use a marginal log-linear parameterisation, under which the model is defined through suitable zero-constraints on the interaction parameters calculated within marginal distributions. We undertake a comprehensive Bayesian analysis of these models, involving suitable choices of prior distributions, estimation, model determination, as well as the allied computational issues. The methodology is illustrated with reference to a real data set. 1 BACKGROUND AND P RELIMINARIES Graphical models of marginal independence were originally introduced by Cox and Wermuth (1993) for the analysis of multivariate Gaussian distributions. They compose a family of mul- tivariate distributions incorporating the marginal independences represented by a bidirected graph. The nodes in the graph correspond to a set of random variables and the bidirected edges represent the pairwise associations between them. A missing edge from a pair of nodes indi- cates that the corresponding variables are marginally independent. The list of independences implied by a bidirected graph can be obtained using the Global Markov property (Kauer- mann, 1996 and Richardson, 2003). The distribution of a random vector X V = {X v , v ∈ V } satisfies the Global Markov property if A is separated from B by V \ (A ∪ B ∪ C) in G implies X A ⊥⊥X B |X C , with A, B and C disjoint subsets of V , and C may be empty. From the global Markov property, we directly derive that if two nodes i and j are disconnected then X i ⊥⊥X j , that is the variables are marginal independent. In the Gaussian case marginal independences correspond to zero constraints in the varian- ce-covariance matrix. The situation is more complicated in the discrete case, where marginal independences correspond to non linear constraints on the set of parameters. Only recently parameterisations for these models have been proposed by Lupparelli et al. (2009) and Dr- ton and Richardson (2008). According to Drton and Richardson (2008), a discrete graphical model of marginal independence, associated to a bidirected graph G, is a family P(G) of joint distributions for a categorical random vector X V satisfying the global Markov property .