Abstract
In this paper, we describe an abstract framework and axioms under which exact local computation of marginals is possible. The primitive objects of the framework are variables and valuations. The primitive operators of the framework are combination and marginalization. These operate on valuations. We state three axioms for these operators and we derive the possibility of local computation from the axioms. Next, we describe a propagation scheme for computing marginals of a valuation when we have a factorization of the valuation on a hypertree. Finally we show how the problem of computing marginals of joint probability distributions and joint belief functions fits the general framework.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnborg, S., Corneil, D. G., and Proskurowski, A. Complexity of finding embeddings in a k-tree. SIAM Journal of Algebraic and Discrete Mathematics 8 (1987), 277–284.
Berge, C. Graphs and Hypergraphs. translated from French by E. Minieka, North-Holland, 1973.
Bertele, U., and Brioschi, F. Nonserial Dynamic Programming. Academic Press, New York, NY, 1972.
Brownston, L. S., Farrell, R. G., Kant, E., and Martin, N. Programming Expert Systems in OPS5: An Introduction to Rule-Based Programming. Addison-Wesley, Reading, MA, 1985. 20 Axioms for Probability and Belief-Function Propagation 527
Cannings, C., Thompson, E. A., and Skolnick, M. H. Probability functions on complex pedigrees. Advances in Applied Probability 10 (1978).
Darroch, J. N., Lauritzen, S. L., and Speed, T. P. Markov fields and log-linear interaction models for contingency tables. The Annals of Statistics 8 (1980), 522–539.
Davis, R., and King, J. J. The origin of rule-based systems in AI. In Rule- Based Expert Systems: The MYCIN Experiments of the Stanford Heuristic Programming Project, B. G. Buchanan and E. H. Shortliffe, Eds. Addison-Wesley, Reading, MA, 1984, pp. 20–52.
Dempster, A. P. New methods for reasoning towards posterior distributions based on sample data. Annals of Mathematical Statistics 37 (1966), 355–374.
Dempster, A. P. Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38 (1967), 325–339.
Dempster, A. P. Uncertain evidence and artificial analysis. Tech. rep., Department of Statistics, Harvard University, Cambridge, MA, 1986.
Golumbic, M. C. Algorithmic Graph Theory and Perfect Graphs. Academic Press, 1980.
Hsia, Y., and Shenoy, P. P. An evidential language for expert systems. In Methodologies for Intelligent Systems, Z. Ras, Ed., vol. 4. North-Holland, 1989.
Hsia, Y., and Shenoy, P. P. MacEvidence: A visual evidential language for knowledge-based systems. Tech. rep., School of Business, University of Kansas, Lawrence, KS, 1989.
Kelly, C. W. I., and Barclay, S. A general Bayesian model for hierarchical inference. Organizational Behavior and Human Performance 10 (1973), 388–403.
Kong, A. Multivariate Belief Functions and Graphical Models. PhD thesis, Department of Statistics, Harvard University, Cambridge, MA, 1986.
Lauritzen, S. L., Speed, T. P., and Vijayan, K. Decomposable graphs and hypergraphs. Journal of Australian Mathematical Society 36, Series A (1984), 12–29.
Lauritzen, S. L., and Spiegelhalter, D. J. Local computations with probabilities on graphical structures and their application to expert systems (with discussion). Journal of the Royal Statistical Society Series B 50 (1988), 157–224.
Maier, D. The Theory of Relational Databases. Computer Science Press, Rockville, 1983.
Mellouli, K. On the Propagation of Beliefs in Networks using the Dempster- Shafer Theory of Evidence. PhD thesis, School of Business, University of Kansas, Lawrence, KS, 1987.
Pearl, J. Fusion, propagation and structuring in belief networks. Artificial Intelligence 29 (1986), 241–288.
Rose, D. J. Triangulated graphs and the elimination process. Journal of Mathematical Analysis and Application 32 (1970), 597–609.
Shachter, R. D., and Heckerman, D. A backwards view for assessment. AI Magazine 8(3) (1986), 55–61.
Shafer, G. A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ, 1976.
Shafer, G., and Logan, R. Implementing Dempster’s rule for hierarchical evidence. Artificial Intelligence 33 (1987), 271–298.
Shafer, G., and Shenoy, P. P. Local computation in hypertrees. Tech. rep., School of Business, University of Kansas, Lawrence, KS, 1988.
Shafer, G., Shenoy, P. P., and Mellouli, K. Propagating belief functions in qualitative Markov trees. International Journal of Approximate Reasoning 3 (1987), 383–411.
Shafer, G., Shenoy, P. P., and Srivastava, R. P. Auditor’s assistant: A knowledge engineering tool for audit decisions. In Auditing Symposium IX: Proceedings of the 1988 Touche Ross University of Kansas Symposium on Auditing Problems (1988), pp. 61–84.
Shenoy, P. P. On Spohn’s rule for revision of beliefs. Tech. rep., School of Business, University of Kansas, Lawrence, KS, 1989.
Shenoy, P. P. A valuation-based language for expert systems. International Journal for Approximate Reasoning 3, 5 (1989), 383–411.
Shenoy, P. P., and Shafer, G. Propagating belief functions using local computations. IEEE Expert 1, 3 (1986), 43–52.
Shenoy, P. P., and Shafer, G. Axioms for discrete optimization using local computations, working paper no. 207. Tech. rep., School of Business, University of Kansas, Lawrence, KS, 1988.
Shenoy, P. P., and Shafer, G. Constraint propagation. Tech. rep., School of Business, University of Kansas, Lawrence, KS, 1988.
Shenoy, P. P., Shafer, G., and Mellouli, K. Propagation of belief functions: A distributed approach. Uncertainty in Artificial Intelligence 2 (1988), 325–336.
Spohn, W. A general non-prbabilistic theory of inductive reasoing. In Uncertainty in Artificial Intellige. nce 4, R. D. Shachter, T. S. Levitt, J. F. Lemmer, and L. N. Kanal, Eds. North-Holland, Amsterdam, 1988, pp. 149–158.
Spohn, W. Ordinal conditional functions: A dynamic theory of epistemic states. In Causation in Decision, Belief Change, and Statistics 2, W. L. Harper and B. Skyrms, Eds. Reidel, Holland, 1988, pp. 105–134.
Tarjan, R. E., and Yannakakis, M. Simple linear time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acylic hypergraphs. SIAM Journal of Computing 13 (1984), 566–579.
Zarley, D. K. An evidential reasoning system. Tech. rep., School of Business, University of Kansas, Lawrance, KS, 1988.
Zarley, D. K., Hsia, Y., and Shafer, G. Evidential reasoning using delief. In Proceedings of the Seventh National Conference on Artificial Intelligence (AAAI-88) (Minneapolis, MN, 1988), vol. 1, pp. 205–209.
Zhang, L. Studies on finding hypertree covers for hypergraphs. Tech. rep., School of Business, University of Kansas, Lawrance, KS, 1988.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Shenoy, P.P., Shafer, G. (2008). Axioms for Probability and Belief-Function Propagation. In: Yager, R.R., Liu, L. (eds) Classic Works of the Dempster-Shafer Theory of Belief Functions. Studies in Fuzziness and Soft Computing, vol 219. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44792-4_20
Download citation
DOI: https://doi.org/10.1007/978-3-540-44792-4_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25381-5
Online ISBN: 978-3-540-44792-4
eBook Packages: EngineeringEngineering (R0)