On Marginalization, Collapsibility and Precollapsibility

Abstract

It is shown that for every undirected graph G over a finite set N and for every nonempty T ⊂ N there exists an undirected graph G ^T over T, called the marginal graph of G for T, such that the class of marginal distributions for T of (discrete) G-Markovian distributions coincides with the class of G ^T-Markovian distributions. An example shows that this is not true within the framework of strictly positive probability distributions. However, an analogous positive result holds for hypergraphs and classes of strictly positive factorizable distributions.