Measuring Inconsistency in Probabilistic Knowledge Bases

This paper develops an inconsistency measure on conditional probabilistic knowledge bases. The measure is based on fundamental principles for inconsistency measures and thus provides a solid theoretical framework for the treatment of inconsistencies in probabilistic expert systems. We illustrate its usefulness and immediate application on several examples and present some formal results. Building on this measure we use the Shapley value---a well-known solution for coalition games---to define a sophisticated indicator that is not only able to measure inconsistencies but to reveal the causes of inconsistencies in the knowledge base. Altogether these tools guide the knowledge engineer in his aim to restore consistency and therefore enable him to build a consistent and usable knowledge base that can be employed in probabilistic expert systems.

[1]  N. Rescher,et al.  On inference from inconsistent premisses , 1970 .

[2]  Didier Dubois,et al.  Some Syntactic Approaches to the Handling of Inconsistent Knowledge Bases: A Comparative Study Part 1: The Flat Case , 1997, Stud Logica.

[3]  A. Hunter,et al.  Shapley Inconsistency Values , 2006, KR.

[4]  Didier Dubois,et al.  Inconsistency in possibilistic knowledge bases: to live with it or not live with it , 1992 .

[5]  Gabriele Kern-Isberner,et al.  Conditionals in Nonmonotonic Reasoning and Belief Revision , 2001, Lecture Notes in Computer Science.

[6]  Didier Dubois,et al.  Possibilistic and Standard Probabilistic Semantics of Conditional Knowledge Bases , 1997, J. Log. Comput..

[7]  Rudolf Kruse,et al.  Fusion: General concepts and characteristics , 2001, Int. J. Intell. Syst..

[8]  Philippe Besnard,et al.  Paraconsistent Reasoning as an Analytic Tool , 2001, Log. J. IGPL.

[9]  Eliezer L. Lozinskii,et al.  Resolving contradictions: A plausible semantics for inconsistent systems , 1994, Journal of Automated Reasoning.

[10]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[11]  Trevor J. M. Bench-Capon,et al.  Argumentation in artificial intelligence , 2007, Artif. Intell..

[12]  Daniele Pretolani,et al.  Easy Cases of Probabilistic Satisfiability , 2001, Annals of Mathematics and Artificial Intelligence.

[13]  Peter Gärdenfors,et al.  On the logic of theory change: Partial meet contraction and revision functions , 1985, Journal of Symbolic Logic.

[14]  Wilhelm Rödder,et al.  Coherent Knowledge Processing at Maximum Entropy by SPIRIT , 1996, UAI.

[15]  Sébastien Konieczny,et al.  Reasoning under inconsistency: the forgotten connective , 2005, IJCAI.

[16]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[17]  Wilhelm Rödder,et al.  Conditional logic and the Principle of Entropy , 2000, Artif. Intell..

[18]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[19]  Anthony Hunter,et al.  Approaches to Measuring Inconsistent Information , 2005, Inconsistency Tolerance.

[20]  W. Spears.  Probabilistic Satisfiability , 1992 .

[21]  L. Shapley.  A Value for n-person Games , 1988 .

[22]  Joseph Y. Halpern,et al.  Random Worlds and Maximum Entropy , 1994, J. Artif. Intell. Res..

[23]  Gabriele Kern-Isberner,et al.  Conditionals in Nonmonotonic Reasoning and Belief Revision: Considering Conditionals as Agents , 2001 .

[24]  Gabriele Kern-Isberner,et al.  Belief Revision and Information Fusion in a Probabilistic Environment , 2003, FLAIRS.

[25]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[26]  Gabriele Kern-Isberner,et al.  Resolving Inconsistencies in Probabilistic Knowledge Bases , 2007, KI.

[27]  Kevin Knight,et al.  Measuring Inconsistency , 2002, J. Philos. Log..

[28]  David Billington.  Propositional Clausal Defeasible Logic , 2008, JELIA.

[29]  J. Paris.  The Uncertain Reasoner's Companion: A Mathematical Perspective , 1994 .