Abstract

: This note is concerned with the class of hierarchical interaction models for mixed discrete and continuous variables as defined by Edwards (1990) and modified by Lauritzen (1996). In particular it is shown that any hierarchical log-linear interaction model can be generated by selection on a set of response variables in a directed Markov model over what we have termed the selection graph of the model. An inequality is established for the entries in the concentration matrix of any Gaussian undirected Markov distribution obtained by conditioning on the values of the response variables in the selection graph, thus demonstrating that not all such distributions can be generated in this way. Finally it is shown that in the mixed case only hierarchical models of the type defined by Edwards (1990) can be generated by selection as above. KEYWORDS: Bayesian networks; Conditional Gaussian distribution; Covariance selection; Gaussian graphical models; Log-linear interaction models; Recursive mode...

Citations