Abstract

By the marginal problem is understood the problem of the existence of a global (full-dimensional) knowledge representation which has prescribed less-dimensional representations as marginals. It arises in several calculi of artificial intelligence: probabilistic reasoning, theory of relational databases, possibility theory, Dempster-Shafer's theory of belief functions, Spohn's theory of ordinal conditional functions. The following result, already known in probabilistic framework and in the framework of relational databases, is shown also for the other calculi: the running intersection property is a necessary and sufficient condition for pairwise compatibility of prescribed less-dimensional knowledge representations being equivalent to the existence of a global representation having them as marginals. Moreover, a simple method of solving the marginal problem in the possibilistic framework and its subframeworks is described.

AMS classification 68T30, 94D05, 68P15

Keywords

marginal problem

database relation

natural conditional function

possibility theory

Dempster-Shafer's theory

running intersection property

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The paper builds on the following works:

• P.P. Shenoy: Conditional independence in valuation-based systems. International Journal of Approximate Reasoning 10 (1994), n. 3, pp. 203-234.
• M. Studeny: Conditional independence and natural conditional functions. International Journal of Approximate Reasoning 12 (1995), n. 1, pp. 43-68.
• H.G. Kellerer: Verteilungsfunktionen mit gegebenem Marginalverteilungen. (in German) Z. Wahrsch. Verw. Gebiete 3 (1964), pp. 247-270.