Probabilistic compositional models: Solution of an equivalence problem

Probabilistic compositional models, similarly to graphical Markov models, are able to represent multidimensional probability distributions using factorization and closely related concept of conditional independence. Compositional models represent an algebraic alternative to the graphical models. The system of related conditional independencies is not encoded explicitly (e.g. using a graph) but it is hidden in a model structure itself. This paper provides answers to the question how to recognize whether two different compositional model structures are equivalent i.e. whether they induce the same system of conditional independencies. Above that, it provides an easy way to convert one structure into an equivalent one in terms of some elementary operations on structures, closely related ability to generate all structures equivalent with a given one, and a unique representative of a class of equivalent structures.