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36
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 20 (8 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 18 (14 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.
Independent natural extension
 IN: IPMU 2010: PROCEEDINGS OF THE 13TH INFORMATION PROCESSING AND MANAGEMENT OF UNCERTAINTY IN KNOWLEDGEBASED SYSTEMS CONFERENCE
, 2010
"... We introduce a general definition for the independence of a number of finitevalued variables, based on coherent lower previsions. Our definition has an epistemic flavour: it arises from personal judgements that a number of variables are irrelevant to one another. We show that a number of already ..."
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Cited by 18 (9 self)
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We introduce a general definition for the independence of a number of finitevalued variables, based on coherent lower previsions. Our definition has an epistemic flavour: it arises from personal judgements that a number of variables are irrelevant to one another. We show that a number of already existing notions, such as strong independence, satisfy our definition. Moreover, there always is a leastcommittal independent model, for which we provide an explicit formula: the independent natural extension. Our central result is that the independent natural extension satisfies socalled marginalisation, associativity and strong factorisation properties. These allow us to relate our research to more traditional ways of defining independence based on factorisation.
Conglomerable Natural Extension
 7TH INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS, INNSBRUCK, AUSTRIA
, 2011
"... We study the weakest conglomerable model that is implied by desirability or probability assessments: the conglomerable natural extension. We show that taking the natural extension of the assessments while imposing conglomerability—the procedure adopted in Walley’s theory—does not yield, in general, ..."
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Cited by 10 (8 self)
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We study the weakest conglomerable model that is implied by desirability or probability assessments: the conglomerable natural extension. We show that taking the natural extension of the assessments while imposing conglomerability—the procedure adopted in Walley’s theory—does not yield, in general, the conglomerable natural extension (but it does so in the case of the marginal extension). Iterating this process produces a sequence of models that approach the conglomerable natural extension, although it is not known, at this point, whether it is attained in the limit. We give sufficient conditions for this to happen in some special cases, and study the differences between working with coherent sets of desirable gambles and coherent lower previsions. Our results indicate that it might be necessary to rethink the foundations of Walley’s theory of coherent conditional lower previsions for infinite partitions of conditioning events.
Updating coherent previsions on finite spaces
 FUZZY SETS AND SYSTEMS
, 2009
"... We compare the different notions of conditional coherence within the behavioural theory of imprecise probabilities when all the spaces are finite. We show that the differences between the notions are due to conditioning on sets of (lower, and in some cases upper) probability zero. Next, we characte ..."
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Cited by 8 (5 self)
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We compare the different notions of conditional coherence within the behavioural theory of imprecise probabilities when all the spaces are finite. We show that the differences between the notions are due to conditioning on sets of (lower, and in some cases upper) probability zero. Next, we characterise the range of coherent extensions in the finite case, proving that the greatest coherent extensions can always be calculated using the notion of regular extension, and we discuss the extensions of our results to infinite spaces.
COMPUTING EXPECTATIONS WITH CONTINUOUS PBOXES: Univariate Case
, 2009
"... Given an imprecise probabilistic model over a continuous space, computing lower/upper expectations is often computationally hard to achieve, even in simple cases. Because expectations are essential in decision making and risk analysis, tractable methods to compute them are crucial in many applica ..."
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Given an imprecise probabilistic model over a continuous space, computing lower/upper expectations is often computationally hard to achieve, even in simple cases. Because expectations are essential in decision making and risk analysis, tractable methods to compute them are crucial in many applications involving imprecise probabilistic models. We concentrate on pboxes (a simple and popular model), and on the computation of lower expectations of nonmonotone functions. This paper is devoted to the univariate case, that is where only one variable has uncertainty. We propose and compare two approaches: the first using general linear programming, and the second using the fact that pboxes are special cases of random sets. We underline the complementarity of both approaches, as well as the differences.
ON THE CONNECTION BETWEEN PROBABILITY BOXES AND POSSIBILITY MEASURES
"... ABSTRACT. We explore the relationship between possibility measures (supremum preserving normed measures) and pboxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and Aeyels, among others. We start by demonstrating th ..."
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ABSTRACT. We explore the relationship between possibility measures (supremum preserving normed measures) and pboxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and Aeyels, among others. We start by demonstrating that only those pboxes who have 0–1valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a pbox to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a pbox, simply by ordering elements by increasing possibility. Whence, any techniques for pboxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for pboxes. Doing so, we arrive at a new rule of combination for possibility measures, for the independent case. 1.
NOTES ON DESIRABILITY AND CONDITIONAL LOWER PREVISIONS
"... Abstract. We detail the relationship between sets of desirable gambles and conditional lower previsions. The former is one the most general models of uncertainty. The latter corresponds to Walley’s celebrated theory of imprecise probability. We consider two avenues: when a collection of conditional ..."
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Abstract. We detail the relationship between sets of desirable gambles and conditional lower previsions. The former is one the most general models of uncertainty. The latter corresponds to Walley’s celebrated theory of imprecise probability. We consider two avenues: when a collection of conditional lower previsions is derived from a set of desirable gambles, and its converse. In either case, we relate the properties of the derived model with those of the originating one. Our results constitute basic tools to move from one formalism to the other, and thus to take advantage of work done in the two fronts. 1.
Conglomerable coherence
 International Journal of Approximate Reasoning
"... We contrast Williams ’ and Walley’s theories of coherent lower previsions in the light of conglomerability. These are two of the most credited approaches to a behavioural theory of imprecise probability. Conglomerability is the notion that distinguishes them the most: Williams ’ theory does not cons ..."
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Cited by 4 (3 self)
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We contrast Williams ’ and Walley’s theories of coherent lower previsions in the light of conglomerability. These are two of the most credited approaches to a behavioural theory of imprecise probability. Conglomerability is the notion that distinguishes them the most: Williams ’ theory does not consider it, while Walley aims at embedding it in his theory. This question is important, as conglomerability is a major point of disagreement at the foundations of probability, since it was first defined by de Finetti in 1930. We show that Walley’s notion of joint coherence (which is the single axiom of his theory) for conditional lower previsions does not take all the implications of conglomerability into account. Considered also some previous results in the literature, we deduce that Williams ’ theory should be the one to use when conglomerability is not required; for the opposite case, we define the new theory of conglomerably coherent lower previsions, which is arguably the one to use, and of which Walley’s theory can be understood as an approximation. We show that this approximation is exact in two important cases: when all conditioning events have positive lower probability, and when conditioning partitions are nested.