Local Computation Schemes with Partially Ordered Preferences

Abstract. Many computational problems linked to reasoning under uncertainty can be expressed in terms of computing the marginal(s) of the combination of a collection of (local) valuation functions. Shenoy and Shafer showed how such a computation can be performed using only local computations. A major strength of this work is that it is based on an algebraic description: what is proved is the correctness of the local computation algorithm under a few axioms on the algebraic structure. The instantiations of the framework in practice make use of totally ordered scales. The present paper focuses on problems of optimization over partially ordered scales, including problems that do not rely on a semilattice, and examines how they can be cast in the Shafer-Shenoy framework so as to satisfy the axioms for local computation and thus benefit from local computation algorithms. It also provides many examples of preference relations, thus showing that each of the algebraic structures explored here has its own interest.