Bucket Elimination : a Unifying Framework

Probabilistic inference algorithms for belief updating, nding the most probable explanation, the maximum a posteriori hypothesis, and the maximum expected utility are reformulated within the bucket elimination framework. This emphasizes the principles common to many of the algorithms appearing in the probabilistic inference literature and clariies the relationship of such algorithms to nonserial dynamic programming algorithms. A general method for combining conditioning and bucket elimination is also presented. For all the algorithms, bounds on complexity are given as a function of the problem's structure. 1. Overview Bucket elimination is a unifying algorithmic framework that generalizes dynamic programming to accommodate algorithms for many complex problem-solving and reasoning activities, including directional resolution for propo-sitional satissability (Davis and Putnam, 1960), adaptive consistency for constraint satisfaction (Dechter and Pearl, 1987), Fourier and Gaussian elimination for linear equalities and inequalities, and dynamic programming for combinatorial optimization (Bertele and Brioschi, 1972). Here, after presenting the framework, we demonstrate that a number of algorithms for probabilistic inference can also be expressed as bucket-elimination algorithms. The main virtues of the bucket-elimination framework are simplicity and generality. By simplicity, we mean that a complete speciication of 2 R. DECHTER bucket-elimination algorithms is feasible without introducing extensive terminology (e.g., graph concepts such as triangulation and arc-reversal), thus making the algorithms accessible to researchers in diverse areas. More important , the uniformity of the algorithms facilitates understanding, which encourages cross-fertilization and technology transfer between disciplines. Indeed, all bucket-elimination algorithms are similar enough for any improvement to a single algorithm to be applicable to all others expressed in this framework. For example, expressing probabilistic inference algorithms as bucket-elimination methods clariies the former's relationship to dynamic programming and to constraint satisfaction such that the knowledge accumulated in those areas may be utilized in the probabilistic framework. The generality of bucket elimination can be illustrated with an algorithm in the area of deterministic reasoning. Consider the following algorithm for deciding satissability. Given a set of clauses (a clause is a dis-junction of propositional variables or their negations) and an ordering of the propositional variables, d = Q 1 (DR) (Dechter and Rish, 1994), is the core of the well-known Davis-Putnam algorithm for satissability (Davis and Putnam, 1960). The algorithm is described using buckets partitioning the given set of clauses such that all the clauses containing Q i that do not contain any symbol higher in the ordering are placed in the bucket of Q …