We present a new kind of structural Markov property for probabilistic laws on
decomposable graphs, which allows the explicit control of interactions between
cliques, so is capable of encoding some interesting structure. We prove the
equivalence of this property to an exponential family assumption, and discuss
identifiability, modelling, inferential and computational implications.Comment: 10 pages, 3 figures; updated from V1 following journal review, new
more explicit title and added section on inferenc