Learning from data: possibilistic graphical models

One of the major problems of handling imperfect information in knowledge-based systems is finding a computationally appealing description of the available data, which is both economical in using storage and supports efficient reasoning techniques. The existence of such a description strongly rests on whether dependencies among the data items are decomposable into local, more basic dependencies. The relational database model [Maier, 1983; Ullman, 1988; Ullman, 1989] has tackled this problem by storing a database in form of a lossless join decomposition, namely a collection of projections, from which the original relation can be reconstructed. Some recent work in this field concerns structure identification in relational data [Dechter and Pearl, 1992] and the clarification of cross-references to the solution of constraint-satisfaction problems [Gyssens et al., 1994]. Although many problems in artificial intelligence, database theory, graph theory, and operations research are connected with decomposition problems, very few general results have been established so far. In the field of graphical modeling and its widespread application areas in diagnostics, expert systems, KDD systems, planning systems, data analysis, and control, the most advanced deliverables refer to probabilistic graphical models [Whittaker, 1990; Castillo et al., 1997; Lauritzen, 1997]. In addition to efficient evidence propagation techniques [Pearl, 1986; Lauritzen and Spiegelhalter, 1988], there are many deliverables that refer to the induction of Bayesian networks or Markov networks from statistical data. For an overview, see [Buntine, 1994; Madigan and Raftery, 1994; Fisher and Lenz, 1996]. The corresponding algorithms are based on linearity and normality assumptions [Pearl and Wermuth, 19931, the extensive testing of conditional independence relations [Spines and Glymour, 1991; Verma and Pearl, 1992], or Bayesian approaches [Cooper and Herskovits, 1992; Lauritzen et al., 1993]. Some crucial problems of these methods concern their computational complexity, their limited reliability unless the amount of data is enormous, and strong presuppositions like the requirement of an ordering of the nodes and a priori distribution assumptions. In order to overcome the complexity problems some heuristic approaches like, for example, the K2 algorithm [Cooper and Herskovits, 1992] with respect to a appropriate quality measure [Geiger and Heckerman, 1995] or the fusion of conditional independence tests with Bayesian learning [Singh and Valtorta, 1993] have turned out to be unavoidable.