WA qualitative probabilistic network models the probabilistic relationships between its variables by means of signs. Non-monotonic influences have associated an ambiguous sign. These ambiguous signs typically lead to uninformative results upon inference. A non-monotonic influence can, however, be associated with a, more informative, sign that indicates its effect in the current state of the network. To capture this effect, we introduce the concept of situational sign. Furthermore, if the network converts to a state in which all variables that provoke the non-monotonicity have been observed, a non-monotonic influence reduces to a monotonic influence. We study the persistence and propagation of situational signs upon inference and give a method to establish the sign of a reduced influence.

The aim of this research is to develop a reasoning under uncertainty strategy in the context of the Fuzzy Inductive Reasoning (FIR) methodology. FIR emerged from the General Systems Problem Solving developed by G. Klir. It is a data driven methodology based on systems behavior rather than on structural knowledge. It is a very useful tool for both the modeling and the prediction of those systems for which no previous structural knowledge is available. FIR reasoning is based on pattern rules synthesized from the available data. The size of the pattern rule base can be very large making the prediction process quite difficult. In order to reduce the size of the pattern rule base, it is possible to automatically extract classical Sugeno fuzzy rules starting from the set of pattern rules. The Sugeno rule base preserves pattern rules knowledge as much as possible. In this process some information is lost but robustness is considerably increased. In the forecasting process either the pattern rule base or the Sugeno fuzzy rule base can be used. The first option is desirable when the computational resources make it possible to deal with the overall pattern rule base or when the extracted fuzzy rules are not accurate enough due to uncertainty associated to the original data. In the second option, the prediction process is done by means of the classical Sugeno inference system. If the amount of uncertainty associated to the data is small, the predictions obtained using the Sugeno fuzzy rule base will be very accurate. In this paper a mixed pattern/fuzzy rules strategy is proposed to deal with uncertainty in such a way that the best of both perspectives is used. Areas in the data space with a higher level of uncertainty are identified by means of the so-called error models. The prediction process in these areas makes use of a mixed pattern/fuzzy rules scheme, whereas areas identified with a lower level of uncertainty only use the Sugeno fuzzy rule base. The proposed strategy is applied to a real biomedical system, i.e., the central nervous system control of the cardiovascular system.

This paper is directed towards combining Pearl's structural-model approach to causal reasoning with high-level formalisms for reasoning about actions. More precisely, we present a combination of Pearl's structural-model approach with Poole's independent choice logic. We show how probabilistic theories in the independent choice logic can be mapped to probabilistic causal models. This mapping provides the independent choice logic with appealing concepts of causality and explanation from the structural-model approach. We illustrate this along Halpern and Pearl's sophisticated notions of actual cause, explanation, and partial explanation. This mapping also adds first-order modeling capabilities and explicit actions to the structural-model approach.

A fundamental question in causal inference is whether it is possible to reliably infer manipulation effects from observational data. There are a variety of senses of asymptotic reliability in the statistical literature, among which the most commonly discussed frequentist notions are pointwise consistency and uniform consistency. Uniform consistency is in general preferred to pointwise consistency because the former allows us to control the worst case error bounds with a finite sample size. In the sense of pointwise consistency, several reliable causal inference algorithms have been established under the Markov and Faithfulness assumptions [Pearl 2000, Spirtes et al. 2001]. In the sense of uniform consistency, however, reliable causal inference is impossible under the two assumptions when time order is unknown and/or latent confounders are present [Robins et al. 2000]. In this paper we present two natural generalizations of the Faithfulness assumption in the context of structural equation models, under which we show that the typical algorithms in the literature (in some cases with modifications) are uniformly consistent even when the time order is unknown. We also discuss the situation where latent confounders may be present and the sense in which the Faithfulness assumption is a limiting case of the stronger assumptions.

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.

Pearls concept OF a d - connecting path IS one OF the foundations OF the modern theory OF graphical models : the absence OF a d - connecting path IN a DAG indicates that conditional independence will hold IN ANY distribution factorising according TO that graph. IN this paper we show that IN singly - connected Gaussian DAGs it IS possible TO USE the form OF a d - connection TO obtain qualitative information about the strength OF conditional dependence.More precisely, the squared partial correlations BETWEEN two given variables, conditioned ON different subsets may be partially ordered BY examining the relationship BETWEEN the d - connecting path AND the SET OF variables conditioned upon.

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.

The property of perfectness plays an important role in the theory of Bayesian networks. First, the existence of perfect distributions for arbitrary sets of variables and directed acyclic graphs implies that various methods for reading independence from the structure of the graph (e.g., Pearl, 1988; Lauritzen, Dawid, Larsen & Leimer, 1990) are complete. Second, the asymptotic reliability of various search methods is guaranteed under the assumption that the generating distribution is perfect (e.g., Spirtes, Glymour & Scheines, 2000; Chickering & Meek, 2002). We provide a lower-bound on the probability of sampling a non-perfect distribution when using a fixed number of bits to represent the parameters of the Bayesian network. This bound approaches zero exponentially fast as one increases the number of bits used to represent the parameters. This result implies that perfect distributions with fixed-length representations exist. We also provide a lower-bound on the number of bits needed to guarantee that a distribution sampled from a uniform Dirichlet distribution is perfect with probability greater than 1/2. This result is useful for constructing randomized reductions for hardness proofs.

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.

Dependability modeling and evaluation is aimed at investigating that a system performs its function correctly in time. A usual way to achieve a high reliability, is to design redundant systems that contain several replicas of the same subsystem or component. State space methods for dependability analysis may suffer of the state space explosion problem in such a kind of situation. Combinatorial models, on the other hand, require the simplified assumption of statistical independence; however, in case of redundant systems, this does not guarantee a reduced number of modeled elements. In order to provide a more compact system representation, parametric system modeling has been investigated in the literature, in such a way that a set of replicas of a given subsystem is parameterized so that only one representative instance is explicitly included. While modeling aspects can be suitably addressed by these approaches, analytical tools working on parametric characterizations are often more difficult to be defined and the standard approach is to 'unfold' the parametric model, in order to exploit standard analysis algorithms working at the unfolded 'ground' level. Moreover, parameterized combinatorial methods still require the statistical independence assumption. In the present paper we consider the formalism of Parametric Fault Tree (PFT) and we show how it can be related to Probabilistic Horn Abduction (PHA). Since PHA is a framework where both modeling and analysis can be performed in a restricted first-order language, we aim at showing that converting a PFT into a PHA knowledge base will allow an approach to dependability analysis directly exploiting parametric representation. We will show that classical qualitative and quantitative dependability measures can be characterized within PHA. Furthermore, additional modeling aspects (such as noisy gates and local dependencies) as well as additional reliability measures (such as posterior probability analysis) can be naturally addressed by this conversion. A simple example of a multi-processor system with several replicated units is used to illustrate the approach.