The paper deals with optimality issues in connection with updating beliefs in networks. We address two processes: triangulation and construction of junction trees. In the first part, we give a simple algorithm for constructing an optimal junction tree from a triangulated network. In the second part, we argue that any exact method based on local calculations must either be less efficient than the junction tree method, or it has an optimality problem equivalent to that of triangulation. 1 INTRODUCTION The junction tree propagation method (Jensen et al., 1990

A probabilistic expert system provides a graphical representation of a joint probability distribution which enables local computations of probabilities. Dawid (1992) provided a `flow-propagation' algorithm for finding the most probable configuration of the joint distribution in such a system. This paper analyses that algorithm in detail, and shows how it can be combined with a clever partitioning scheme to formulate an efficient method for finding the M most probable configurations. The algorithm is a divide and conquer technique, that iteratively identifies the M most probable configurations. The algorithm has been implemented into the experimental shell XBAIES, which is an extension of BAIES (Cowell, 1992). Keywords: Bayesian network, belief revision, most probable explanation, junction tree, maximization, propagation, charge, potential function, conditional independence, flow, evidence, marginalization, divide-andconquer. 1 Introduction A probabilistic expert system (PES) funct...

We introduce a methodology for performing approximate computations in very complex probabilistic systems (e.g. huge pedigrees). Our approach, called blocking Gibbs, combines exact local computations with Gibbs sampling in a way that complements the strengths of both. The methodology is illustrated on a real-world problem involving a heavily inbred pedigree containing 20;000 individuals. We present results showing that blocking-Gibbs sampling converges much faster than plain Gibbs sampling for very complex problems.

This paper discusses a method for implementing a probabilistic inference system based on an extended relational data model.

An extension of the expert system shell HUGIN to include continuous wriables, in the form of linear additive normally distributed variables, is presented. The

Several scoring metrics are used in different search procedures for learning probabilistic networks. We study the properties of cross entropy in learning a decomposable Markov network. Though entropy and related scoring metrics were widely used, its `microscopic' properties and asymptotic behavior in a search have not been analyzed. We present such a `microscopic' study of a minimum entropy search algorithm, and show that it learns an I-map of the domain model when the data size is large. Search procedures that modify a network structure one link at a time have been commonly used for efficiency. Our study indicates that a class of domain models cannot be learned by such procedures. This suggests that prior knowledge about the problem domain together with a multi-link search strategy would provide an effective way to uncover many domain models.

An Evolution Program is presented to propagate convex sets of probabilities. This algorithm is useful when the number of extreme points in the 'a posteriori' convex set for a variable is too high and a single probabilistic propagation is feasible. We have tested the algorithm in a random causal network with a random number of conditional probabilities in each one of the variables. The experimental evaluation show that the resulting intervals obtained for the cases of a variable are similar to those obtained using an exact method of propagation. 1 Introduction Graphical structures have been used to represent and manipulate independence relationships [16]. Propagation algorithms were first developed for the probabilistic case [11, 17]. These independence relationships can be used to obtain factorizations of uncertainty representations given by means of other formalisms (see Shafer and Shenoy [19, 18]). One of the particularizations of the Shafer and Shenoy system was given by Cano, Mora...

Several recent papers have described probability models which used graph and hypergraphs to represent relationships among the variables. Two related computing algorithms are commonly used to manipulate such models: the peeling algorithm which eliminates variables one at a time to find the marginal distribution of a single variable, and the fusion and propagation algorithm which simultaneously solves for many marginal distributions by passing messages in a Tree of Cliques whose nodes correspond to subsets of variables. The peeling algorithm requires an elimination order. As demonstrated in this paper, the elimination order can in turn be used to construct a Tree of Cliques for propagation and fusion. This paper addresses three computational issues: 1) The choice of elimination order determines the size of the largest node of the Tree of Cliques, which dominates the computational cost for the probability model using either peeling or fusion and propagation. We review heuristics for choosing an elimination order. (2) Inserting intersection nodes into the tree of cliques produces a junction tree which has a lower computational cost. We present an algorithm which produces a junction trees with a high computational efficiency. (3) Augmenting the tree of cliques with additional nodes can lead to a new tree structure which more clearly expresses the relationship between the original graphical model and the tree model.

Dawid, Kjærulff & Lauritzen (1994) provided a preliminary description of a hybrid between Monte-Carlo sampling methods and exact local computations in junction trees. Utilizing the strengths of both methods, such hybrid inference methods has the potential of expanding the class of problems which can be solved under bounded resources as well as solving problems which otherwise resist exact solutions. The paper provides a detailed description of a particular instance of such a hybrid scheme; namely, combination of exact inference and Gibbs sampling in discrete Bayesian networks. We argue that this combination calls for an extension of the usual message passing scheme of ordinary junction trees.

The undirected technique for evaluating belief networks [ Jensen et al., 1990a, Lauritzen and Spiegelhalter, 1988 ] requires clustering the nodes in the network into a junction tree. In the traditional view, the junction tree is constructed from the cliques of the moralized and triangulated belief network: triangulation is taken to be the primitive concept, the goal towards which any clustering algorithm (e.g. node elimination) is directed. In this paper, we present an alternative conception of clustering, in which clusters and the junction tree property play the role of primitives: given a graph (not a tree) of clusters which obey (a modified version of) the junction tree property, we transform this graph until we have obtained a tree. There are several advantages to this approach: it is much clearer and easier to understand, which is important for humans who are constructing belief networks; it admits a wider range of heuristics which may enable more efficient o...