Conditional Independence in Uncertainty Theories

This paper introduces the notions of independence and conditional independence in valuation-based systems (VBS). VBS is an axiomatic framework capable of representing many different uncertainty calculi. We define independence and conditional independence in terms of factorization of the joint valuation. The definitions of independence and conditional independence in VBS generalize the corresponding definitions in probability theory. Our definitions apply not only to probability theory, but also to Dempster-Shafer's belief-function theory, Spohn's epistemic-belief theory, and Zadeh's possibility theory. In fact, they apply to any uncertainty calculi that fit in the framework of valuation-based systems.

[1]  J. Q. Smith.  Influence Diagrams for Statistical Modelling , 1989 .

[2]  A. Dawid.  Conditional Independence in Statistical Theory , 1979 .

[3]  Arie Tzvieli.  Possibility theory: An approach to computerized processing of uncertainty , 1990, J. Am. Soc. Inf. Sci..

[4]  Wolfgang Spohn,et al.  Ordinal Conditional Functions: A Dynamic Theory of Epistemic States , 1988 .

[5]  Judea Pearl,et al.  GRAPHOIDS: A Graph-based logic for reasoning about relevance relations , 1985 .

[6]  Glenn Shafer,et al.  Perspectives on the theory and practice of belief functions , 1990, Int. J. Approx. Reason..

[7]  Wolfgang Spohn,et al.  Stochastic independence, causal independence, and shieldability , 1980, J. Philos. Log..

[8]  Prakash P. Shenoy,et al.  A valuation-based language for expert systems , 1989, Int. J. Approx. Reason..

[9]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

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

[11]  Prakash P. Shenoy,et al.  Valuation-based systems: a framework for managing uncertainty in expert systems , 1992 .

[12]  D. Geiger.  Graphoids: a qualitative framework for probabilistic inference , 1990 .

[13]  Lotfi A. Zadeh,et al.  A Theory of Approximate Reasoning , 1979 .

[14]  Prakash P. Shenoy,et al.  Probability propagation , 1990, Annals of Mathematics and Artificial Intelligence.

[15]  Prakash P. Shenoy,et al.  Axioms for probability and belief-function proagation , 1990, UAI.

[16]  Khaled Mellouli,et al.  Propagating belief functions in qualitative Markov trees , 1987, Int. J. Approx. Reason..

[17]  Daniel Hunter,et al.  Graphoids and natural conditional functions , 1991, Int. J. Approx. Reason..

[18]  Glenn Shafer,et al.  Readings in Uncertain Reasoning , 1990 .

[19]  David C. Hogg,et al.  Advances in Artificial Intelligence-II , 1987 .