It is shown that the ability of the interval probability representation to capture epistemological independence is severely limited. Two events are epistemologically independent if knowledge of the first event does not alter belief (i.e., probability bounds) about the second. However, independence in this form can only exist in a 2-monotone probability function in degenerate cases i.e., if the prior bounds are either point probabilities or entirely vacuous. Additional limitations are characterized for other classes of lower probabilities as well. It is argued that these phenomena are simply a matter of interpretation. They appear to be limitations when one interprets probability bounds as a measure of epistemological indeterminacy (i.e., uncertainty arising from a lack of knowledge), but are exactly as one would expect when probability intervals are interpreted as representations of ontological indeterminacy (indeterminacy introduced by structural approximations). The ontological interpretation is introduced and discussed.

Approaches to learning Bayesian networks from data typically combine a scoring function with a heuristic search procedure. Given a Bayesian network structure, many of the scoring functions derived in the literature return a score for the entire equivalence class to which the structure belongs. When using such a scoring function, it is appropriate for the heuristic search algorithm to search over equivalence classes of Bayesian networks as opposed to individual structures. We present the general formulation of a search space for which the states of the search correspond to equivalence classes of structures. Using this space, any one of a number of heuristic search algorithms can easily be applied. We compare greedy search performance in the proposed search space to greedy search performance in a search space for which the states correspond to individual Bayesian network structures.

The paper presents an efficient method for simulating the tails of a target variable Z=h(X) which depends on a set of basic variables X=(X_1, ..., X_n). To this aim, variables X_i, i=1, ..., n are sequentially simulated in such a manner that Z=h(x_1, ..., x_i-1, X_i, ..., X_n) is guaranteed to be in the tail of Z. When this method is difficult to apply, an alternative method is proposed, which leads to a low rejection proportion of sample values, when compared with the Monte Carlo method. Both methods are shown to be very useful to perform a sensitivity analysis of Bayesian networks, when very large confidence intervals for the marginal/conditional probabilities are required, as in reliability or risk analysis. The methods are shown to behave best when all scores coincide. The required modifications for this to occur are discussed. The methods are illustrated with several examples and one example of application to a real case is used to illustrate the whole process.

Bayesian networks provide a language for qualitatively representing the conditional independence properties of a distribution. This allows a natural and compact representation of the distribution, eases knowledge acquisition, and supports effective inference algorithms. It is well-known, however, that there are certain independencies that we cannot capture qualitatively within the Bayesian network structure: independencies that hold only in certain contexts, i.e., given a specific assignment of values to certain variables. In this paper, we propose a formal notion of context-specific independence (CSI), based on regularities in the conditional probability tables (CPTs) at a node. We present a technique, analogous to (and based on) d-separation, for determining when such independence holds in a given network. We then focus on a particular qualitative representation scheme - tree-structured CPTs - for capturing CSI. We suggest ways in which this representation can be used to support effective inference algorithms. In particular, we present a structural decomposition of the resulting network which can improve the performance of clustering algorithms, and an alternative algorithm based on cutset conditioning.

An algorithm is developed for finding a close to optimal junction tree of a given graph G. The algorithm has a worst case complexity O(c^k n^a) where a and c are constants, n is the number of vertices, and k is the size of the largest clique in a junction tree of G in which this size is minimized. The algorithm guarantees that the logarithm of the size of the state space of the heaviest clique in the junction tree produced is less than a constant factor off the optimal value. When k = O(log n), our algorithm yields a polynomial inference algorithm for Bayesian networks.

Graphical Markov models use graphs, either undirected, directed, or mixed, to represent possible dependences among statistical variables. Applications of undirected graphs (UDGs) include models for spatial dependence and image analysis, while acyclic directed graphs (ADGs), which are especially convenient for statistical analysis, arise in such fields as genetics and psychometrics and as models for expert systems and Bayesian belief networks. Lauritzen, Wermuth and Frydenberg (LWF) introduced a Markov property for chain graphs, which are mixed graphs that can be used to represent simultaneously both causal and associative dependencies and which include both UDGs and ADGs as special cases. In this paper an alternative Markov property (AMP) for chain graphs is introduced, which in some ways is a more direct extension of the ADG Markov property than is the LWF property for chain graph.

We developed the language of Modifiable Temporal Belief Networks (MTBNs) as a structural and temporal extension of Bayesian Belief Networks (BNs) to facilitate normative temporal and causal modeling under uncertainty. In this paper we present definitions of the model, its components, and its fundamental properties. We also discuss how to represent various types of temporal knowledge, with an emphasis on hybrid temporal-explicit time modeling, dynamic structures, avoiding causal temporal inconsistencies, and dealing with models that involve simultaneously actions (decisions) and causal and non-causal associations. We examine the relationships among BNs, Modifiable Belief Networks, and MTBNs with a single temporal granularity, and suggest areas of application suitable to each one of them.

We extend Gaussian networks - directed acyclic graphs that encode probabilistic relationships between variables - to its vector form. Vector Gaussian continuous networks consist of composite nodes representing multivariates, that take continuous values. These vector or composite nodes can represent correlations between parents, as opposed to conventional univariate nodes. We derive rules for inference in these networks based on two methods: message propagation and topology transformation. These two approaches lead to the development of algorithms, that can be implemented in either a centralized or a decentralized manner. The domain of application of these networks are monitoring and estimation problems. This new representation along with the rules for inference developed here can be used to derive current Bayesian algorithms such as the Kalman filter, and provide a rich foundation to develop new algorithms. We illustrate this process by deriving the decentralized form of the Kalman filter. This work unifies concepts from artificial intelligence and modern control theory.

The criterion commonly used in directed acyclic graphs (dags) for testing graphical independence is the well-known d-separation criterion. It allows us to build graphical representations of dependency models (usually probabilistic dependency models) in the form of belief networks, which make easy interpretation and management of independence relationships possible, without reference to numerical parameters (conditional probabilities). In this paper, we study the following combinatorial problem: finding the minimum d-separating set for two nodes in a dag. This set would represent the minimum information (in the sense of minimum number of variables) necessary to prevent these two nodes from influencing each other. The solution to this basic problem and some of its extensions can be useful in several ways, as we shall see later. Our solution is based on a two-step process: first, we reduce the original problem to the simpler one of finding a minimum separating set in an undirected graph, and second, we develop an algorithm for solving it.

Although probabilistic inference in a general Bayesian belief network is an NP-hard problem, computation time for inference can be reduced in most practical cases by exploiting domain knowledge and by making approximations in the knowledge representation. In this paper we introduce the property of similarity of states and a new method for approximate knowledge representation and inference which is based on this property. We define two or more states of a node to be similar when the ratio of their probabilities, the likelihood ratio, does not depend on the instantiations of the other nodes in the network. We show that the similarity of states exposes redundancies in the joint probability distribution which can be exploited to reduce the computation time of probabilistic inference in networks with multiple similar states, and that the computational complexity in the networks with exponentially many similar states might be polynomial. We demonstrate our ideas on the example of a BN2O network -- a two layer network often used in diagnostic problems -- by reducing it to a very close network with multiple similar states. We show that the answers to practical queries converge very fast to the answers obtained with the original network. The maximum error is as low as 5% for models that require only 10% of the computation time needed by the original BN2O model.