Inverses, Conditionals and Compositional Operators in Separative Valuation Algebra

Compositional models were introduce by Jirousek and Shenoy in the general framework of valuation-based systems. They based their theory on an axiomatic system of valuations involving not only the operations of combination and marginalisation, but also of removal. They claimed that this systems covers besides the classical case of discrete probability distributions, also the cases of Gaussian densities and belief functions, and many other systems. Whereas their results on the compositional operator are correct, the axiomatic basis is not sufficient to cover the examples claimed above. We propose here a different axiomatic system of valuation algebras, which permits a rigorous mathematical theory of compositional operators in valuation-based systems and covers all the examples mentioned above. It extends the classical theory of inverses in semigroup theory and places thereby the present theory into its proper mathematical frame. Also this theory sheds light on the different structures of valuation-based systems, like regular algebras (represented by probability potentials), canncellative algebras (Gaussian potentials) and general separative algebras (density functions).