Pairwise Markov properties for regression graphs

With sequences of regressions, one may generate joint probability distributions. One starts with a marginal distribution of context variables of concentration graph structure and continues with ordered sequences of conditional distributions, named regressions in joint responses. The involved random variables may be discrete, continuous or of both type. Such a generating process determines for each response the conditioning set containing its regressor variables and an ordering of the node set in the corresponding regression graph with three types of edge. This graph contains different possible types of pairwise Markov properties, which interpret the conditional independence associated with a missing edge in the graph. We explain how these properties arise and prove their equivalence for compositional graphoids.