This paper presents new results on the complexity of graph-theoretical models that represent probabilities (Bayesian networks) and that represent interval and set valued probabilities (credal networks). We define a new class of networks with bounded width, and introduce a new decision problem for Bayesian networks, the maximin a posteriori. We present new links between the Bayesian and credal networks, and present new results both for Bayesian networks (most probable explanation with observations, maximin a posteriori) and for credal networks (bounds on probabilities a posteriori, most probable explanation with and without observations, maximum a posteriori). 1

Introduction Reasoning in terms of cause and effect is a strategy that arises in many tasks. For example, diagnosis is usually defined as the task of finding the causes (illnesses) from the observed effects (symptoms). Similarly, prediction can be understood as the description of a future plausible situation where observed effects will be in accordance with the known causal structure of the phenomenon being studied. Causal models are a summary of the knowledge about a phenomenon expressed in terms of causation. Many areas of the ap- # This work has been partially supported by the Spanish Comission Interministerial de Ciencia y Tecnologia Project CICYT-TIC96 -0878. plied sciences (econometry, biomedics, engineering, etc.) have used such a term to refer to models that yield explanations, allow for prediction and facilitate planning and decision making. Causal reasoning can be viewed as inference guided by a causation theory. That kind of inference can be further specialised into induc

Robust Bayesian inference is the calculation of posterior probability bounds given perturbations in a probabilistic model. This paper focuses on perturbations that can be expressed locally in Bayesian networks through convex sets of distributions. Two approaches for combination of local models are considered. The rst approach takes the largest set of joint distributions that is compatible with the local sets of distributions; we show how to reduce this type of robust inference to a linear programming problem. The second approach takes the convex hull of joint distributions generated from the local sets of distributions; we demonstrate how to apply interior-point optimization methods to generate posterior bounds and how to generate approximations that are guaranteed to converge to correct posterior bounds. We also discuss calculation of bounds for expected utilities and variances, and global perturbation models. 1

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Different uncertainty propagation algorithms in graphical structures can be viewed as a particular case of propagation in a joint tree, which can be obtained from different triangulations of the original graph. The complexity of the resulting propagation algorithms depends on the size of the resu lting triangulated graph. The prob lem of obtaining an optimum graph triangu lation is known to be NP-complete. Thus approximate algorithms which find a good triangulation in reasonable time are of particular interest. This work describes and compares several heuristic algorithms developed for this purpose.

A credal network associates sets of probability distributions with directed acyclic graphs. Under strong independence assumptions, inference with credal networks is equivalent to a signomial program under linear constraints, a problem that is NP-hard even for categorical variables and polytree models. We describe

Modelling a decision support system for

This paper investigates a representation language with flexibility inspired by probabilistic logic and compactness inspired by relational Bayesian networks. The goal is to handle propositional and first-order constructs together with precise, imprecise, indeterminate and qualitative probabilistic assessments. The paper shows how this can be achieved through the theory of credal networks. New exact and approximate inference algorithms based on multilinear programming and iterated/loopy propagation of interval probabilities are presented; their superior performance, compared to existing ones, is shown empirically. 1

This paper analyzes irrelevance and independence relations in graphical models associated with convex sets of probability distributions (called Quasi-Bayesian networks). The basic question in Quasi-Bayesian networks is, How can irrelevance/independence relations in Quasi-Bayesian networks be detected, enforced and exploited? This paper addresses these questions through Walley's definitions of irrelevance and independence. Novel algorithms and results are presented for inferences with the so-called natural extensions using fractional linear programming, and the properties of the so-called type-1 extensions are clarified through a new generalization of d-separation. 1

Kuznetsov's condition says that variables X and Y are independent when any product of bounded functions f (X) and g(Y ) behaves in a certain way: the interval of expected values f (X)g(Y )] must be equal to the interval product f (X)]E[g(Y)]. The main result of this paper shows how to compute lower expectations using Kuznetsov's condition. We also generalize Kuznetsov's condition to conditional expectation intervals, and study the relationship between Kuznetsov's conditional condition and the semi-graphoid properties.