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72
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 32 (10 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 28 (7 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
Graphoid properties of epistemic irrelevance and independence
, 2005
"... This paper investigates Walley’s concepts of epistemic irrelevance and epistemic independence for imprecise probability models. We study the mathematical properties of irrelevance and independence, and their relation to the graphoid axioms. Examples are given to show that epistemic irrelevance can v ..."
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Cited by 27 (3 self)
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This paper investigates Walley’s concepts of epistemic irrelevance and epistemic independence for imprecise probability models. We study the mathematical properties of irrelevance and independence, and their relation to the graphoid axioms. Examples are given to show that epistemic irrelevance can violate the symmetry, contraction and intersection axioms, that epistemic independence can violate contraction and intersection, and that this accords with informal notions of irrelevance and independence.
Probabilistic logic and probabilistic networks
, 2008
"... While in principle probabilistic logics might be applied to solve a range of problems, in practice they are rarely applied at present. This is perhaps because they seem disparate, complicated, and computationally intractable. However, we shall argue in this programmatic paper that several approaches ..."
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Cited by 17 (13 self)
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While in principle probabilistic logics might be applied to solve a range of problems, in practice they are rarely applied at present. This is perhaps because they seem disparate, complicated, and computationally intractable. However, we shall argue in this programmatic paper that several approaches to probabilistic logic fit into a simple unifying framework: logically complex evidence can be used to associate probability intervals or probabilities with sentences. Specifically, we show in Part I that there is a natural way to present a question posed in probabilistic logic, and that various inferential procedures provide semantics for that question: the standard probabilistic semantics (which takes probability functions as models), probabilistic argumentation (which considers the probability of a hypothesis being a logical consequence of the available evidence), evidential probability (which handles reference classes and frequency data), classical statistical inference
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
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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
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 (6 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.
Conditional Plausibility Measures and Bayesian Networks
, 2001
"... A general notion of algebraic conditional plausibility measures is de#ned. Probability measures, ranking functions, possibility measures, and #under the appropriate de#nitions# sets of probability measures can all be viewed as de#ning algebraic conditional plausibility measures. It is shown that alg ..."
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Cited by 9 (4 self)
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A general notion of algebraic conditional plausibility measures is de#ned. Probability measures, ranking functions, possibility measures, and #under the appropriate de#nitions# sets of probability measures can all be viewed as de#ning algebraic conditional plausibility measures. It is shown that algebraic conditional plausibility measures can be represented using Bayesian networks. 1.
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