Computing Marginals Using Local Computation

This paper describes an abstract framework called valuation network for computation of marginals using local computation. In valuation networks, we represent knowledge using functions called valuations. Making inferences involves using two operators called marginalization and combination. Marginalization tells us how to coarsen a valuation by eliminating some variables. Combination tells us how to combine valuations. Making inferences from a valuation network can be simply described as finding the marginal of the joint valuation for each variable of interest. The joint valuation is the combination of all valuations. We state some simple axioms that marginalization and combination need to satisfy to enable us to compute marginals using local computation. We describe a fusion algorithm for computing a marginal using local computation. We describe the fusion algorithm as a message passing in join tree algorithm. Next we describe how the messagepassing algorithm can be adapted to compute multiple marginals efficiently. Finally, we describe how Bayesian probability theory, Dempster-Shafer's belief function theory, Spohn's epistemic belief theory, and Zadeh's possibility theory, constraint satisfaction problems, and various other problems fit in the VN framework.