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27
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.
SYMMETRY OF MODELS VERSUS MODELS OF SYMMETRY
, 2008
"... A model for a subject’s beliefs about a phenomenon may exhibit symmetry, in the sense that it is invariant under certain transformations. On the other hand, such a belief model may be intended to represent that the subject believes or knows that the phenomenon under study exhibits symmetry. We def ..."
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Cited by 12 (6 self)
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A model for a subject’s beliefs about a phenomenon may exhibit symmetry, in the sense that it is invariant under certain transformations. On the other hand, such a belief model may be intended to represent that the subject believes or knows that the phenomenon under study exhibits symmetry. We defend the view that these are fundamentally different things, even though the difference cannot be captured by Bayesian belief models. In fact, the failure to distinguish between both situations leads to Laplace’s socalled Principle of Insufficient Reason, which has been criticised extensively in the literature. We show that there are belief models (imprecise probability models, coherent lower previsions) that generalise and include the Bayesian belief models, but where this fundamental difference can be captured. This leads to two notions of symmetry for such belief models: weak invariance (representing symmetry of beliefs) and strong invariance (modelling beliefs of symmetry). We discuss various mathematical as well as more philosophical aspects of these notions. We also discuss a few examples to show the relevance of our findings both to probabilistic modelling and to statistical inference, and to the notion of exchangeability in particular.
Cooman. Marginal extension in the theory of coherent lower previsions
 International Journal of Approximate Reasoning
, 2007
"... ABSTRACT. We generalise Walley’s Marginal Extension Theorem to the case of any finite number of conditional lower previsions. Unlike the procedure of natural extension, our marginal extension always provides the smallest (most conservative) coherent extensions. We show that they can also be calculat ..."
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Cited by 12 (11 self)
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ABSTRACT. We generalise Walley’s Marginal Extension Theorem to the case of any finite number of conditional lower previsions. Unlike the procedure of natural extension, our marginal extension always provides the smallest (most conservative) coherent extensions. We show that they can also be calculated as lower envelopes of marginal extensions of conditional linear (precise) previsions. Finally, we use our version of the theorem to study the socalled forward irrelevant product and forward irrelevant natural extension of a number of marginal lower previsions. 1.
Conservative inference rule for uncertain reasoning under incompleteness
 Journal of Artificial Intelligence Research
"... In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process ’ behaviour to be partly unknown. Then we use Walley’s theory ..."
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Cited by 10 (7 self)
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In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process ’ behaviour to be partly unknown. Then we use Walley’s theory of coherent lower previsions, a generalisation of the Bayesian theory to imprecision, to derive the rule to update beliefs under incompleteness that logically follows from our assumptions, and that we call conservative inference rule. This rule has some remarkable properties: it is an abstract rule to update beliefs that can be applied in any situation or domain; it gives us the opportunity to be neither too optimistic nor too pessimistic about the incompleteness process, which is a necessary condition to draw reliable while strong enough conclusions; and it is a coherent rule, in the sense that it cannot lead to inconsistencies. We give examples to show how the new rule can be applied in expert systems, in parametric statistical inference, and in pattern classification, and discuss more generally the view of incompleteness processes defended here as well as some of its consequences. 1.
Epistemic irrelevance in credal nets: the case of imprecise Markov trees
, 2010
"... We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability ..."
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Cited by 9 (8 self)
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We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact messagepassing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithm’s operation, and report an application to online character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.
IMPRECISE PROBABILITY TREES: BRIDGING TWO THEORIES OF IMPRECISE PROBABILITY
, 2008
"... We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley’s behavioural theory of imprecise probabilities, and Shafer and Vovk’s gametheoretic account of probability. We show that the two theories are more closely re ..."
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Cited by 9 (4 self)
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We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley’s behavioural theory of imprecise probabilities, and Shafer and Vovk’s gametheoretic account of probability. We show that the two theories are more closely related than would be suspected at first sight, and we establish a correspondence between them that (i) has an interesting interpretation, and (ii) allows us to freely import results from one theory into the other. Our approach leads to an account of probability trees and random processes in the framework of Walley’s theory. We indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees, and we prove an interesting and quite general version of the weak law of large numbers.
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
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.
Decision making under incomplete data using the imprecise Dirichlet model
, 2006
"... The paper presents an efficient solution to decision problems where direct partial information on the distribution of the states of nature is available, either by observations of previous repetitions of the decision problem or by direct expert judgements. To process this information we use a recent ..."
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Cited by 5 (2 self)
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The paper presents an efficient solution to decision problems where direct partial information on the distribution of the states of nature is available, either by observations of previous repetitions of the decision problem or by direct expert judgements. To process this information we use a recent generalization of Walley’s imprecise Dirichlet model, allowing us also to handle incomplete observations or imprecise judgements. We derive efficient algorithms and discuss properties of the optimal solutions. In the case of precise data and pure actions we are surprisingly led to a frequencybased variant of the HodgesLehmann criterion, which was developed in classical decision theory as a compromise between Bayesian and minimax procedures.