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Probability Intervals Over Influence Diagrams
March 1993 (vol. 15 no. 3)
pp. 280-286

A mechanism for performing probabilistic reasoning in influence diagrams using interval rather than point-valued probabilities is described. Procedures for operations corresponding to conditional expectation and Bayesian conditioning in influence diagrams are derived where lower bounds on probabilities are stored at each node. The resulting bounds for the transformed diagram are shown to be the tightest possible within the class of constraints on probability distributions that can be expressed exclusively as lower bounds on the component probabilities of the diagram. Sequences of these operations can be performed to answer probabilistic queries with indeterminacies in the input and for performing sensitivity analysis on an influence diagram. The storage requirements and computational complexity of this approach are comparable to those for point-valued probabilistic inference mechanisms.

[1] J. S. Breese and K. W. Fertig, "Decision making with interval influence diagrams," inProc. Sixth Conf. Uncertainty Artificial Intell.(Cambridge, MA), Aug. 1990, pp. 122-129.
[2] P. Chu, H. Moskowitz, and R. Wong, "Robust interactive decision-analysis (RID): Concepts, methodology, and system principles," Tech. Rep., Krannert Grad. Sch. Manag., Purdue Univ., West Lafayette, IN, May 1988.
[3] G. F. Cooper, "NESTOR: A computer-based medical diagnostic aid that integrates causal and probabilistic knowledge," Ph.D. thesis, Comput. Sci. Dept., Stanford Univ., Nov 1984. Rep. no. STAN-CS-84-48; Also numbered HPP-84-48.
[4] K. W. Fertig and J. S. Breese, "Probability intervals over influence diagrams," Tech. Rep., Rockwell Int. Sci. Cent., Rockwell Res. Rep. 4, Mar. 1990.
[5] D. Geiger, T. Verma, and J. Pearl, "Identifying independence in Bayesian networks,"Networks, vol. 20, pp. 507-534, 1990.
[6] M. L. Ginsberg, "Does probability have a place in nonmonotonic reasoning," inProc. Ninth Int. Joint Conf. Artificial Intell.(Los Angeles), 1985, pp. 447-449.
[7] B. Grosof, "An inequality paradigm for probabilistic knowledge: The logic of conditional probability intervals," inUncertainty in Artificial Intelligence(L. N. Kanal and J. F. Lemmer, Eds.). Amsterdam: North Holland, 1986, pp. 259-275.
[8] P. Haddawy, "Implementation and experiments with a variable precision logic system," inProc. AAAI-86 Fifth Nat. Conf. Artificial Intell(Philadelphia), Aug. 1986, pp. 238-242.
[9] R. A. Howard and J. E. Matheson, "Influence diagrams, inReadings on the Principles and Applications of Decision Analysis(R. A. Howard and J. E. Matheson, Eds.). Menlo Park, CA: Strategic Decisions Group, 1981, pp. 721-762, vol. II.
[10] H. Kyburg, "Bayesian and non-Bayesian evidential updating,"Artificial Intell., vol. 31, pp. 271-293, 1987.
[11] H. Kyburg, "Higher order probabilities and intervals,"Int. J. Approximate Reasoning, vol. 2, pp. 195-208, 1988.
[12] I. Levi,The Enterprise of Knowledge. Cambridge, MA: MIT Press, 1980.
[13] R. Loui, "Theory and computation of uncertain inference," Ph.D. thesis, Dept. Comput. Sci., Univ. of Rochester, 1987; also available as TR-228, Univ. of Rochester, Dept. of Comput. Sci.
[14] R. F. Nau, "Decision analysis with indeterminate or incoherent probabilities,"Annals Oper. Res., vol. 19, pp. 375-403, 1989.
[15] R. E. Neapolitan and J. Kenevan, "Justifying the principle of interval constraints, inProc. Amer. Assoc. Artificial Intell. Fourth Workshop Uncertainty(Minneapolis, MN), Aug. 1988.
[16] S. M. Olmsted, "On representing and solving decision problems," Ph.D. thesis, Dept. of Eng.-Econ. Syst., Stanford Univ., Dec. 1983.
[17] J. Pearl, "On probability intervals,"Int. J. Approximate Reasoning, vol. 2, pp. 211-216, 1988.
[18] T. Seidenfeld, M. Schervish, and J. Kadane, "Decisions without ordering," Tech. Rep. 391, Dept. of Stat., Carnegie Mellon Univ., Pittsburgh, Mar. 1987.
[19] R. D. Shachter, "Evaluating influence diagrams,"Oper. Res., vol. 34, no. 6, pp. 871-882, 1986; reprinted in [42].
[20] R. D. Shachter, "Probabilistic inference and influence diagrams,"Oper. Res., vol. 36, pp. 589-604, 1988.
[21] P. Snow, "Bayesian inference without point estimates," inProc. Amer. Assoc. Artificial Intell. Fifth Nat. Conf.(Philadelphia), Aug. 1986, pp. 233-237.
[22] L. van der Gaag, "Computing probability intervals under independency constraints," inProc. Sixth Conf. Uncertainty Artificial Intell.(Cambridge, MA), Aug. 1990.
[23] M. P. Wellman, "Fundamental concepts of qualitative probabilistic networks,"Artificial Intell., vol. 44, pp. 257-303, 1990.
[24] C. C. White,A posteriorirepresentations based on linear inequality descriptions ofa prioriand conditional probabilities,"IEEE Trans. Syst. Man Cybern., vol. 16, no. 4, pp. 570-573, 1986.

Index Terms:
influence diagrams; probabilistic reasoning; conditional expectation; Bayesian conditioning; lower bounds; probability distributions; probabilistic queries; sensitivity analysis; computational complexity; point-valued probabilistic inference mechanisms; Bayes methods; inference mechanisms; probability; sensitivity analysis; uncertainty handling
K.W. Fertig, J.S. Breese, "Probability Intervals Over Influence Diagrams," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 3, pp. 280-286, March 1993, doi:10.1109/34.204910
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