This paper provides conditions and procedures for deciding if patterns of independencies found in covariance and concentration matrices can be generated by a stepwise recursive process represented by some directed acyclic graph. If such an agreement is found, we know that one or several causal processes could be responsible for the observed independencies, and our procedures could then be used to elucidate the graphical structure common to these processes, so as to evaluate their compatibility against substantive knowledge of the domain. If we find that the observed pattern of independencies does not agree with any stepwise recursive process, then there are a number of different possibilities. For instance, -- some weak dependencies could have been mistaken for independencies and led to the wrong omission of edges from the covariance or concentration graphs. -- some of the observed linear dependencies reflect accidental cancellations or hide actual nonlinear relations, or -- the process responsible for the data is non-recursive, involving aggregated variables, simultenous reciprocal interactions, or mixtures of several causal processes. In order to recognize accidental independencies it would be helpful to conduct several longitudinal studies under slightly varying conditions. In such studies the covariances for the same set of variables is estimated under different conditions and the variations in the conditions would typically affect the numerical values of the parameters. But, if the data were generated by a causal process represented by some directed acyclic graph, then the basic structural properties reflected in the missing edges of that graph should remain unchanged. Under such assumptions, the pattern of independencies that is "implied" by the dag (see Definitio...

In Bayesian analysis of multi-way contingency tables, the selection of a prior distribution for either the log-linear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical log-linear models, which includes the class of graphical models. These priors are defined as the Diaconis–Ylvisaker conjugate priors on the log-linear parameters subject to “baseline constraints ” under multinomial sampling. We also derive the induced prior on the cell probabilities and show that the induced prior is a generalization of the hyper Dirichlet prior. We show that this prior has several desirable properties and illustrate its usefulness by identifying the most probable decomposable, graphical and hierarchical log-linear models for a six-way contingency table. 1. Introduction. We

Abstract. We discuss two parameterizations of models for marginal independencies for discrete distributions which are representable by bi-directed graph models, under the global Markov property. Such models are useful data analytic tools especially if used in combination with other graphical models. The first parameterization, in the saturated case, is also known as the multivariate logistic transformation, the second is a variant that allows, in some (but not all) cases, variation independent parameters. An algorithm for maximum likelihood fitting is proposed, based on an extension of the Aitchison and Silvey method.

type for categorical data