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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 15 (8 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
"... ..."
Belief updating and learning in semiqualitative probabilistic networks
 Conference on Uncertainty in Artificial Intelligence. AUAI
, 2005
"... This paper explores semiqualitative probabilistic networks (SQPNs) that combine numeric and qualitative information. We first show that exact inferences with SQPNs are NP PPComplete. We then show that existing qualitative relations in SQPNs (plus probabilistic logic and imprecise assessments) can ..."
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Cited by 9 (5 self)
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This paper explores semiqualitative probabilistic networks (SQPNs) that combine numeric and qualitative information. We first show that exact inferences with SQPNs are NP PPComplete. We then show that existing qualitative relations in SQPNs (plus probabilistic logic and imprecise assessments) can be dealt effectively through multilinear programming. We then discuss learning: we consider a maximum likelihood method that generates point estimates given a SQPN and empirical data, and we describe a Bayesianminded method that employs the Imprecise Dirichlet Model to generate setvalued estimates. 1
Equivalence between Bayesian and credal nets on an updating problem
 Soft Methods for Integrated Uncertainty Modeling
, 2006
"... We establish an intimate connection between Bayesian and credal nets. Bayesian nets are precise graphical models, credal nets extend Bayesian nets to imprecise probability. We focus on traditional belief updating with credal nets, and on the kind of belief updating that arises with Bayesian nets whe ..."
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Cited by 8 (7 self)
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We establish an intimate connection between Bayesian and credal nets. Bayesian nets are precise graphical models, credal nets extend Bayesian nets to imprecise probability. We focus on traditional belief updating with credal nets, and on the kind of belief updating that arises with Bayesian nets when the reason for the missingness of some of the unobserved variables in the net is unknown. We show that the two updating problems are formally the same. 1
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.
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 5 (2 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.
Databases for Interval Probabilities
, 2004
"... We present a database framework for the efficient storage and manipulation of interval probability distributions and their associated information. While work on interval probabilities and on probabilistic databases has appeared before, ours is the first to combine these into a coherent and mathemati ..."
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Cited by 5 (1 self)
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We present a database framework for the efficient storage and manipulation of interval probability distributions and their associated information. While work on interval probabilities and on probabilistic databases has appeared before, ours is the first to combine these into a coherent and mathematically sound framework including both standard relational queries and queries based on probability theory. In particular, our query algebra allows users not only to query existing interval probability distributions, but also to construct new ones by means of conditionalization and marginalization, as well as other more common database
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 4 (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].
Efficient Solutions to Factored MDPs with Imprecise Transition Probabilities
"... When modeling realworld decisiontheoretic planning problems in the Markov decision process (MDP) framework, it is often impossible to obtain a completely accurate estimate of transition probabilities. For example, natural uncertainty arises in the transition specification due to elicitation of MDP ..."
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Cited by 3 (0 self)
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When modeling realworld decisiontheoretic planning problems in the Markov decision process (MDP) framework, it is often impossible to obtain a completely accurate estimate of transition probabilities. For example, natural uncertainty arises in the transition specification due to elicitation of MDP transition models from an expert or data, or nonstationary transition distributions arising from insufficient state knowledge. In the interest of obtaining the most robust policy under transition uncertainty, the Markov Decision Process with Imprecise Transition Probabilities (MDPIPs) has been introduced to model such scenarios. Unfortunately, while solutions to the MDPIP are wellknown, they require nonlinear optimization and are extremely timeconsuming in practice. To address this deficiency, we propose efficient dynamic programming methods to exploit the structure of factored MDPIPs. Noting that the key computational bottleneck in the solution of MDPIPs is the need to repeatedly solve nonlinear constrained optimization problems, we show how to target approximation techniques to drastically reduce the computational overhead of the nonlinear solver while producing bounded, approximately optimal solutions. Our results show up to two orders of magnitude speedup in comparison to traditional “flat ” dynamic programming approaches and up to an order of magnitude speedup over the extension of factored MDP approximate value iteration techniques to MDPIPs.