This paper discusses a method for im plementing a probabilistic inference system based on an extended relational data model. This model provides a unified approach for a variety of applications such as dynamic pro gramming, solving sparse linear equations, and constraint propagation. In this frame work, the probability model is represented as a generalized relational database. Subse quent probabilistic requests can be processed as standard relational queries. Conventional database management systems can be easily adopted for implementing such an approxi mate reasoning system. Probabilistic models [4, 9, 10] are used for making de cisions under uncertainty. The input to a probabilistic model is usually a Bayesian network [10] . It may also consist of a set of potentials which define a Markov network [4] . In this paper, we assume that the proba bilistic model is described by a Markov network. For this model, the propagation method [5, 6, 7, 12, 13] can be conveniently applied to convert the potentials into marginal distributions. There is another important reason to characterize a probabilistic model by a Markov network, as it has been shown that such a network can be represented as a generalized relational database (14, 15, 16] . That is, the probabilistic model can be transformed into an equivalent (extended) relational data model. More specifically, the marginal corresponding to each po tential can be viewed as a relation in the relational database. Furthermore, the database scheme derived from a Markov network forms an acyclic join depen dency [15] , which possesses many desirable properties [1, 8] in database applications. As the probabilistic model is now represented by a re lational data model, a probability request expressed as a conditional probability can be equivalently trans formed into a standard query to be executed by the database management system. Naturally, all query optimization techniques can be directly applied to pro cessing this query including data structure modifica tion. Thus, these transformations allow us to take full advantage of the query optimizer and other per formance enhancement capabilities available in tradi tional relational databases. This paper, a sequel of the presentation in the IPMU conference [15] , reports on the technical details in volved in the design of a probabilistic inference system by transforming a Markov network into a relational database. Our paper is organized as follows. In Section 2, for completeness we review a unified relational data model for both probabilistic reasoning and database manage ment systems. In Section 3, we show that a factored probability distribution can be expressed as a general ized acyclic join dependency. The method for imple menting a probabilistic inference system is described in Section 4. First, we describe how a relational database is constructed for a given probabilistic model. We then show that processing a request for evidential reason ing is equivalent to processing a standard relational query. We conclude by pointing out that the extended relational database system can in fact model a number of apparently different but closely related applications [12] . Before introducing our data model, we need to define some basic notions pertinent to our discussion such as: hypergraphs, factored distributions, and marginal ization. Then we show how under certain conditions a factored joint probability distribution can be expressed
This translation of the clique functions of a Markov net to a generalised relational database is essentially the same as that of REF .
A Method for Implementing a Probabilistic Model as a Relational Database
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