The two most popular types of graphical model are directed models (Bayesian networks) and undirected models (Markov random fields, or MRFs). Directed and undirected models offer complementary properties in model construction, expressing conditional independencies, expressing arbitrary factorizations of joint distributions, and formulating message-passing inference algorithms. We show that the strengths of these two representations can be combined in a single type of graphical model called a 'factor graph'. Every Bayesian network or MRF can be easily converted to a factor graph that expresses the same conditional independencies, expresses the same factorization of the joint distribution, and can be used for probabilistic inference through application of a single, simple message-passing algorithm. In contrast to chain graphs, where message-passing is implemented on a hypergraph, message-passing can be directly implemented on the factor graph. We describe a modified 'Bayes-ball' algorithm for establishing conditional independence in factor graphs, and we show that factor graphs form a strict superset of Bayesian networks and MRFs. In particular, we give an example of a commonly-used 'mixture of experts' model fragment, whose independencies cannot be represented in a Bayesian network or an MRF, but can be represented in a factor graph. We finish by giving examples of real-world problems that are not well suited to representation in Bayesian networks and MRFs, but are well-suited to representation in factor graphs.

Identifying tractable subclasses and designing efficient algorithms for these tractable classes are important topics in the study of constraint satisfaction and Bayesian network inference problems. In this paper we investigate the asymptotic average behavior of a typical tractable subclass characterized by the treewidth of the problems. We show that the property of having a bounded treewidth in the constraint satisfaction problem and Bayesian network inference problem has a phase transition that occurs while the underlying structures of problems are still sparse. This implies that algorithms making use of treewidth based structural knowledge only work efficiently in a limited range of random instances. INTRODUCTION It is well known that many NP complete problems have tractable subclasses characterized by certain structural parameters.

This paper studies decision making for Walley's partially consonant belief functions (pcb). In a pcb, the set of foci are partitioned. Within each partition, the foci are nested. The pcb class includes probability functions and possibility functions as extreme cases. Unlike earlier proposals for a decision theory with belief functions, we employ an axiomatic approach. We adopt an axiom system similar in spirit to von Neumann - Morgenstern's linear utility theory for a preference relation on pcb lotteries. We prove a representation theorem for this relation. Utility for a pcb lottery is a combination of linear utility for probabilistic lottery and binary utility for possibilistic lottery.

In this paper, we introduce a method for approximating the solution to inference and optimization tasks in uncertain and deterministic reasoning. Such tasks are in general intractable for exact algorithms because of the large number of dependency relationships in their structure. Our method effectively maps such a dense problem to a sparser one which is in some sense "closest". Exact methods can be run on the sparser problem to derive bounds on the original answer, which can be quite sharp. We present empirical results demonstrating that our method works well on the tasks of belief inference and finding the probability of the most probable explanation in belief networks, and finding the cost of the solution that violates the smallest number of constraints in constraint satisfaction problems. On one large CPCS network, for example, we were able to calculate upper and lower bounds on the conditional probability of a variable, given evidence, that were almost identical in the average case.

By elaborating on the notion of linear belief functions (Dempster 1990; Liu 1996), we propose an elementary approach to knowledge representation for expert systems using linear belief functions. We show how to use basic matrices to represent market information and financial knowledge, including complete ignorance, statistical observations, subjective speculations, distributional assumptions, linear relations, and empirical asset pricing models. We then appeal to Dempster's rule of combination to integrate the knowledge for assessing an overall belief of portfolio performance, and updating the belief by incorporating additional information. We use an example of three gold stocks to illustrate the approach.

In this paper, we provide new complexity results for algorithms that learn discrete-variable Bayesian networks from data. Our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest model able to represent the generative distribution exactly. Our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufficiently large dataset. We show that identifying high-scoring structures is hard, even when we are given an independence oracle, an inference oracle, and/or an information oracle. Our negative results also apply to the learning of discrete-variable Bayesian networks in which each node has at most k parents, for all k > 3.

Constraint-based (CB) learning is a formalism for learning a causal network with a database D by performing a series of conditional-independence tests to infer structural information. This paper considers a new test of independence that combines ideas from Bayesian learning, Bayesian network inference, and classical hypothesis testing to produce a more reliable and robust test. The new test can be calculated in the same asymptotic time and space required for the standard tests such as the chi-squared test, but it allows the specification of a prior distribution over parameters and can be used when the database is incomplete. We prove that the test is correct, and we demonstrate empirically that, when used with a CB causal discovery algorithm with noninformative priors, it recovers structural features more reliably and it produces networks with smaller KL-Divergence, especially as the number of nodes increases or the number of records decreases. Another benefit is the dramatic reduction in the probability that a CB algorithm will stall during the search, providing a remedy for an annoying problem plaguing CB learning when the database is small.

In Non - ergodic belief networks the posterior belief OF many queries given evidence may become zero.The paper shows that WHEN belief propagation IS applied iteratively OVER arbitrary networks(the so called, iterative OR loopy belief propagation(IBP)) it IS identical TO an arc - consistency algorithm relative TO zero - belief queries(namely assessing zero posterior probabilities). This implies that zero - belief conclusions derived BY belief propagation converge AND are sound.More importantly it suggests that the inference power OF IBP IS AS strong AND AS weak, AS that OF arc - consistency.This allows the synthesis OF belief networks FOR which belief propagation IS useless ON one hand, AND focuses the investigation OF classes OF belief network FOR which belief propagation may be zero - complete.Finally, ALL the above conclusions apply also TO Generalized belief propagation algorithms that extend loopy belief propagation AND allow a crisper understanding OF their power.

Symbolic representations have been used successfully in off-line planning algorithms for Markov decision processes. We show that they can also improve the performance of on-line planners. In addition to reducing computation time, symbolic generalization can reduce the amount of costly real-world interactions required for convergence. We introduce Symbolic Real-Time Dynamic Programming (or sRTDP), an extension of RTDP. After each step of on-line interaction with an environment, sRTDP uses symbolic model-checking techniques to generalizes its experience by updating a group of states rather than a single state. We examine two heuristic approaches to dynamic grouping of states and show that they accelerate the planning process significantly in terms of both CPU time and the number of steps of interaction with the environment.

Most methods of exact probability propagation in Bayesian networks do not carry out the inference directly over the network, but over a secondary structure known as a junction tree or a join tree (JT). The process of obtaining a JT is usually termed {sl compilation}. As compilation is usually viewed as a whole process; each time the network is modified, a new compilation process has to be carried out. The possibility of reusing an already existing JT, in order to obtain the new one regarding only the modifications in the network has received only little attention in the literature. In this paper we present a method for incremental compilation of a Bayesian network, following the classical scheme in which triangulation plays the key role. In order to perform incremental compilation we propose to recompile only those parts of the JT which can have been affected by the networks modifications. To do so, we exploit the technique OF maximal prime subgraph decomposition in determining the minimal subgraph(s) that have to be recompiled, and thereby the minimal subtree(s) of the JT that should be replaced by new subtree(s).We focus on structural modifications : addition and deletion of links and variables.