On association models defined over independence graphs

In the recent extensive research on the use of independence or Markov graphs to study associations and dependences in multivariate systems, each variable is represented by a node; some pairs of nodes are joined by edges, sometimes directed and sometimes undirected and a key role is played by conditional independence. The absence of an edge between two nodes means that the corresponding variables are conditionally independent, the conditioning set depending on the nature of the graph. See Lauritzen and Wermuth (1989), Cox and Wermuth (1993, 1996), Edwards (1995), Lauritzen (1996) and Wermuth (1997). For a description of various kinds of special dependence which may be so represented. The following qualitative distinction is important in applications but rather difficult to capture formally in probabilistic theory. In a statistical model corresponding to a given graph an edge that is present typically corresponds to a free parameter, e.g. a correlation or regression coefficient, which may take any value in the relevant parameter space, including values at zero, the value for a particular independence. On the other hand in a substantive research hypothesis (Wermuth and Lauritzen 1990) an edge that is present is to represent a dependence large enough to be of subject-matter interest. Now the magnitude of such an effect depends on the context. All that we can require for a general discussion is to characterize situations in which rules for specifying and reading off a given graph conditional independences and conditional associations do not overlook some of these