Compositional models, Bayesian models and recursive factorization models

summary:Compositional models are used to construct probability distributions from lower-order probability distributions. On the other hand, Bayesian models are used to represent probability distributions that factorize according to acyclic digraphs. We introduce a class of models, called \emph{recursive factorization models}, to represent probability distributions that recursively factorize according to sequences of sets of variables, and prove that they have the same representation power as both compositional models generated by sequential expressions and Bayesian models. Moreover, we present a linear (graphical) algorithm for deciding if a conditional independence is valid in a given recursive factorization model