Local computation of Gaussian belief functions

AbstractGaussian belief functions represent logical and probabilistic knowledge for mixed variables, some of which are deterministic, some vacuous, and some Gaussian. They include as special types linear equations, statistical observations, multivariate Gaussian distributions, and vacuous belief functions. The notion of Gaussian belief functions was proposed by A.P. Dempster (Normal belief functions and the Kalman filter, Technical report, Department of Statistics, Harvard University, Cambridge, MA, 1990.), formalized by G. Shafer (A note on Dempster's Gaussian belief functions, Technical report, School of Business, University of Kansas, Lawrence, KS, 1992.) and L. Liu (International Journal of Approximate Reasoning 14 (1996) 95–126.); (in: D. Fisher, Hans-J. Lenz (Eds.), Learning Models from Data: AI and Statistics V, Springer, New York, NY, 1996, pp. 79–88.) and successfully applied in combining independent statistical models in L. Liu (Model combination using Gaussian belief functions, Technical report, School of Business, University of Kansas, Lawrence, KS, 1995.). In this paper, we propose a join-tree computation scheme for expert systems using Gaussian belief functions. We first represent Dempster's rule of combination obtained in Liu (1996) alternatively in terms of matrix sweepings. We then show the operations of Gaussian belief functions follow the axioms of P.P. Shenoy and G. Shafer (in: R.D. Shachter, T.S. Levitt, L.N. Kanal, J.F. Lemmer (Eds.), Uncertainty in Artificial Intelligence, vol. 4, North-Holland, Amsterdam, 1990, pp. 169–198.) and justify the possibility of a join-tree computation scheme for Gaussian Belief functions. The result enriches the theory of local computation by extending its applicability to the combination of statistical models and the integration of knowledge bases. Examples are carried out to illustrate how combined inference can be made in accordance with multiple statistical models using graphically structured belief function models