Abstract
In the previous chapter Information Algebra, an algebraic structure capturing the idea that pieces of information refer to precise questions and that they can be combined and focussed on other questions is presented and discussed. A prototype of such information algebras is relational algebra. But also various kind of logic systems induce information algebras. In this chapter, this framework will be used to study uncertain information. It is often the case that a piece of information is known to be valid under certain assumptions, but it is not altogether sure that these assumptions really hold. Varying the assumptions leads to different information. Given such an uncertain body of information, assumption-based reasoning permits to deduce certain conclusions or to prove certain hypotheses under some assumptions. This kind of assumption-based inference can be carried further if the varying likelihood of different assumptions is described by a probability measure on the assumptions. Then, it is possible to measure the degree of support of a hypothesis by the probability that the assumptions supporting the hypothesis hold. A prototype system of such a probabilistic argumentation system based on propositional logic is described in [3]. Another example will be described in Section 2 of this chapter. It is shown that this way to model uncertain information leads to a theory which generalizes the well-known Dempster-Shafer theory [6, 19].
Research supported by grant No. 200020–109510 of the Swiss National Foundation for Research.
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References
Anrig, B., Kohlas, J.: Model-based reliability and diagnostic: A common framework for reliability and diagnostics. Int. J. of Intell. Systems 18(10), 1001–1033 (2003)
Dempster, A.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38, 325–339 (1967)
Haenni, R., Kohlas, J., Lehmann, N.: Probabilistic argumentation systems. In: Kohlas, J., Moral, S. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems. Algorithms for Uncertainty and Defeasible Reasoning, vol. 5, pp. 221–287. Kluwer, Dordrecht (2000)
Kappos, D.A.: Probability Algebras and Stochastic Spaces. Academic Press, New York (1969)
Kohlas, J.: Computational theory for information systems. Technical Report 97–07, Institute of Informatics, University of Fribourg (1997)
Kohlas, J.: Information Algebras: Generic Structures for Inference. Discrete Mathematics and Theoretical Computer Science. Springer, Heidelberg (2003)
Kohlas, J.: Probabilistic argumentation systems: A new way to combine logic with probability. Journal of Applied Logic 1(3-4), 225–253 (2003)
Kohlas, J.: Uncertain information: random variables in graded semilattices. Int. J. Approx. Reason.(2007) doi:10.1016/j.ijar.2006.12.005
Kohlas, J., Haenni, R., Lehmann, N.: Assumption-based reasoning and probabilistic argumentation systems. In: Kohlas, J., Moral, S. (eds.) Algorithms for Uncertainty and Defeasible Reasoning (1999) (accepted for Publication)
Kohlas, J., Monney, P.-A.: Statistical Information. Assumption-Based Statistical Inference. Sigma Series in Stochastics, vol. 3. Heldermann, Lemgo (2008)
Kohlas, J., Monney, P.A.: Representation of evidence by hints. In: Yager, R.R., Kacprzyk, J., Fedrizzi, M. (eds.) Advances in the Dempster-Shafer Theory of Evidence, pp. 472–492. John Wiley & Sons, Chichester (1994)
Kohlas, J., Monney, P.A.: A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence. Lecture Notes in Economics and Mathematical Systems, vol. 425. Springer, Heidelberg (1995)
Kohlas, J., Monney, P.A.: A Mathematical Theory of Hints. An Approach to the Dempster-Shafer Theory of Evidence. Lecture Notes in Economics and Mathematical Systems, vol. 425. Springer, Heidelberg (1995)
Kohlas, J., Shenoy, P.P.: Computation in valuation algebras. In: Kohlas, J., Moral, S. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems, Algorithms for Uncertainty and Defeasible Reasoning, vol. 5, pp. 5–39. Kluwer, Dordrecht (2000)
Lauritzen, S.L., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. J. Royal Statis. Soc. B 50, 157–224 (1988)
Molchanov, I.: Theory of Random Sets. Springer, London (2005)
Monney, P.-A.: A Mathematical Theory of Arguments for Statistical Evidence. In: Contributions to Statistics. Physica-Verlag (2003)
Nguyen, H.: On random sets and belief functions. Journal of Mathematical Analysis and Applications 65, 531–542 (1978)
Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)
Shenoy, P.P., Shafer, G.: Axioms for probability and belief-function propagation. In: Ross, D., Shachter, T.S., Levitt, L.N. (eds.) Uncertainty in Artificial Intelligence 4. Machine intelligence and pattern recognition, vol. 9, pp. 169–198. Elsevier, Amsterdam (1990)
Shenoy, P.P., Shafer, G.: Axioms for probability and belief function propagation. In: Lemmer, J.F., Shachter, R.D., Levitt, T.S., Kanal, L.N. (eds.) Uncertainty in Artif. Intell., vol. 4, pp. 169–198. North-Holland, Amsterdam (1990)
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Kohlas, J., Eichenberger, C. (2009). Uncertain Information. In: Sommaruga, G. (eds) Formal Theories of Information. Lecture Notes in Computer Science, vol 5363. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00659-3_6
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