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Updating Beliefs with Incomplete Observations
"... Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete (or setvalued). This is a fundamental problem in general, and of particular interest for Bayesian networks. Recently, Gr unwald and Halpern have shown that co ..."
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Cited by 41 (13 self)
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Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete (or setvalued). This is a fundamental problem in general, and of particular interest for Bayesian networks. Recently, Gr unwald and Halpern have shown that commonly used updating strategies fail in this case, except under very special assumptions. In this paper we propose a new method for updating probabilities with incomplete observations. Our approach is deliberately conservative: we make no assumptions about the socalled incompleteness mechanism that associates complete with incomplete observations. We model our ignorance about this mechanism by a vacuous lower prevision, a tool from the theory of imprecise probabilities, and we use only coherence arguments to turn prior into posterior (updated) probabilities. In general, this new approach to updating produces lower and upper posterior probabilities and previsions (expectations), as well as partially determinate decisions. This is a logical consequence of the existing ignorance about the incompleteness mechanism. As an example, we use the new updating method to properly address the apparent paradox in the `Monty Hall' puzzle. More importantly, we apply it to the problem of classification of new evidence in probabilistic expert systems, where it leads to a new, socalled conservative updating rule.
The inferential complexity of Bayesian and credal networks
 In Proceedings of the International Joint Conference on Artificial Intelligence
, 2005
"... This paper presents new results on the complexity of graphtheoretical 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 Ba ..."
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Cited by 35 (9 self)
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This paper presents new results on the complexity of graphtheoretical 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
Decisiontheoretic specification of credal networks: a unified language for uncertain modeling with sets of Bayesian networks
 International Journal of Approximate Reasoning
"... Credal networks are models that extend Bayesian nets to deal with imprecision in probability, and can actually be regarded as sets of Bayesian nets. Credal nets appear to be powerful means to represent and deal with many important and challenging problems in uncertain reasoning. We give examples to ..."
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Cited by 21 (9 self)
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Credal networks are models that extend Bayesian nets to deal with imprecision in probability, and can actually be regarded as sets of Bayesian nets. Credal nets appear to be powerful means to represent and deal with many important and challenging problems in uncertain reasoning. We give examples to show that some of these problems can only be modeled by credal nets called nonseparately specified. These, however, are still missing a graphical representation language and updating algorithms. The situation is quite the opposite with separately specified credal nets, which have been the subject of much study and algorithmic development. This paper gives two major contributions. First, it delivers a new graphical language to formulate any type of credal network, both separately and nonseparately specified. Second, it shows that any nonseparately specified net represented with the new language can be easily transformed into an equivalent separately specified net, defined over a larger domain. This result opens up a number of new outlooks and concrete outcomes: first of all, it immediately enables the existing algorithms for separately specified credal nets to be applied to nonseparately specified ones. We explore this possibility for the 2U algorithm: an algorithm for exact updating of singly connected credal nets, which is extended by our results to a class of nonseparately specified models. We also consider the problem of inference on Bayesian networks, when the reason that prevents some of the variables from being observed is unknown. The problem is first reformulated in the new graphical language, and then mapped into an equivalent problem on a separately specified net. This provides a first algorithmic approach to this kind of inference, which is also proved to be NPhard by similar transformations based on our formalism.
IPE and L2U: Approximate algorithms for credal networks
 IN PROCEEDINGS OF THE SECOND STARTING AI RESEARCHER SYMPOSIUM
, 2004
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Bayesian Networks with Imprecise Probabilities: Theory and Application to Classification
, 2010
"... Bayesian network are powerful probabilistic graphical models for modelling uncertainty. Among others, classification represents an important application: some of the most used classifiers are based on Bayesian networks. Bayesian networks are precise models: exact numeric values should be provided fo ..."
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Cited by 12 (3 self)
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Bayesian network are powerful probabilistic graphical models for modelling uncertainty. Among others, classification represents an important application: some of the most used classifiers are based on Bayesian networks. Bayesian networks are precise models: exact numeric values should be provided for quantification. This requirement is sometimes too narrow. Sets instead of single distributions can provide a more realistic description in these cases. Bayesian networks can be generalized to cope with sets of distributions. This leads to a novel class of imprecise probabilistic graphical models, called credal networks. In particular, classifiers based on Bayesian networks are generalized to socalled credal classifiers. Unlike Bayesian classifiers, which always detect a single class as the one maximizing the posterior class probability, a credal classifier may eventually be unable to discriminate a single class. In other words, if the available information is not sufficient, credal classifiers allow for indecision between two or more classes, this providing a less informative but more robust conclusion than Bayesian classifiers.
Inference in Credal Networks with BranchAndBound Algorithms
 IN INT. SYMP. ON IMPRECISE PROBABILITIES AND THEIR APPLICATIONS
, 2003
"... 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 NPhard even for categorical variables and polytree mo ..."
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Cited by 11 (1 self)
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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 NPhard even for categorical variables and polytree models. We describe
Propositional and relational Bayesian networks associated with imprecise and qualitative probabilistic assessments
 IN PROCEEDINGS OF THE 20TH ANNUAL CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE
, 2004
"... 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 firstorder constructs together with precise, imprecise, indeterminate and qualitative probabilistic as ..."
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Cited by 8 (4 self)
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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 firstorder 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.
Locally specified credal networks
 IN PROCEEDINGS OF THE THIRD EUROPEAN WORKSHOP ON PROBABILISTIC GRAPHICAL MODELS (PGM2006
, 2006
"... Credal networks are models that extend Bayesian nets to deal with imprecision in probability, and can actually be regarded as sets of Bayesian nets. Evidence suggests that credal nets are a powerful means to represent and deal with many important and challenging problems in uncertain reasoning. We g ..."
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Cited by 6 (2 self)
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Credal networks are models that extend Bayesian nets to deal with imprecision in probability, and can actually be regarded as sets of Bayesian nets. Evidence suggests that credal nets are a powerful means to represent and deal with many important and challenging problems in uncertain reasoning. We give examples to show that some of these problems can only be modelled by credal nets called nonseparately specified. These, however, are still missing a graphical representation language and solution algorithms. The situation is quite the opposite with separately specified credal nets, which have been the subject of much study and algorithmic development. This paper gives two major contributions. First, it delivers a new graphical language to formulate any type of credal network, both separately and nonseparately specified. Second, it shows that any nonseparately specified net represented with the new language can be easily transformed into an equivalent separately specified net, defined over a larger domain. This result opens up a number of new perspectives and concrete outcomes: first of all, it immediately enables the existing algorithms for separately specified credal nets to be applied to nonseparately specified ones.
Credal nets with probabilities estimated with an extreme imprecise dirichlet model
 PROCEEDINGS OF THE FIFTH INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS (ISIPTA ’07), ACTION M AGENCY
, 2007
"... The propagation of probabilities in credal networks when probabilities are estimated with a global imprecise Dirichlet model is an important open problem. Only Zaffalon [21] has proposed an algorithm for the Naive classifier. The main difficulty is that, in general, computing upper and lower probabi ..."
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Cited by 5 (0 self)
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The propagation of probabilities in credal networks when probabilities are estimated with a global imprecise Dirichlet model is an important open problem. Only Zaffalon [21] has proposed an algorithm for the Naive classifier. The main difficulty is that, in general, computing upper and lower probability intervals implies the resolution of an optimization of a fraction of two polynomials. In the case of the Naive credal classifier, Zaffalon has shown that the function is a convex function of only one parameter, but there is not a similar result for general credal sets. In this paper, we propose the use of an imprecise global model, but we restrict the distributions to only the most extreme ones. The result is a model giving rise that in the case of estimating a conditional probability under independence relationships, it can produce smaller intervals than the global general model. Its main advantage is that the optimization problem is simpler, and available procedures can be directly applied, as the ones proposed in [7].
Assessing Debris Flow Hazard by Credal Nets
 SOFT METHODOLOGY AND RANDOM INFORMATION SYSTEMS (PROCEEDINGS OF THE SECOND INTERNATIONAL CONFERENCE ON SOFT METHODS IN PROBABILITY AND STATISTICS
, 2004
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