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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.
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.
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 8 (5 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
"... Abstract. At the foundations of probability theory lies a question that has been open since de Finetti framed it in 1930: whether or not an uncertainty model should be required to be conglomerable. Conglomerability is related to accepting infinitely many conditional bets. Walley is one of the author ..."
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Cited by 4 (3 self)
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Abstract. At the foundations of probability theory lies a question that has been open since de Finetti framed it in 1930: whether or not an uncertainty model should be required to be conglomerable. Conglomerability is related to accepting infinitely many conditional bets. Walley is one of the authors who have argued in favor of conglomerability, while de Finetti rejected the idea. In this paper we study the extension of the conglomerability condition to two types of uncertainty models that are more general than the ones envisaged by de Finetti: sets of desirable gambles and coherent lower previsions. We focus in particular on the weakest (i.e., the leastcommittal) of those extensions, which we call the conglomerable natural extension. The weakest extension that does not take conglomerability into account is simply called the natural extension. We show that taking the natural extension of assessments after 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 of imposing conglomerability and taking the natural extension produces a sequence of models that approach the conglomerable natural extension, although it is not known, at this point, whether this sequence converges to it. 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 is necessary to rethink the foundations of Walley’s theory of coherent lower previsions for infinite partitions of conditioning events. 1.
CONDITIONAL MODELS: COHERENCE AND INFERENCE THROUGH SEQUENCES OF JOINT MASS FUNCTIONS
"... Abstract. We call a conditional model any set of statements made of conditional probabilities or expectations. We take conditional models as primitive compared to unconditional probability, in the sense that conditional statements do not need to be derived from an unconditional probability. We focus ..."
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Cited by 2 (2 self)
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Abstract. We call a conditional model any set of statements made of conditional probabilities or expectations. We take conditional models as primitive compared to unconditional probability, in the sense that conditional statements do not need to be derived from an unconditional probability. We focus on two problems: (coherence) giving conditions to guarantee that a conditional model is selfconsistent; (inference) delivering methods to derive new probabilistic statements from a selfconsistent conditional model. We address these problems in the case where the probabilistic statements can be specified imprecisely through sets of probabilities, while restricting the attention to finite spaces of possibilities. Using Walley’s theory of coherent lower previsions, we fully characterise the question of coherence, and specialise it for the case of precisely specified probabilities, which is the most common case addressed in the literature. This shows that coherent conditional models are equivalent to sequences of (possibly sets of) unconditional mass functions. In turn, this implies that the inferences from a conditional model are the limits of the conditional inferences obtained by applying Bayes ’ rule, when possible, to the elements of the sequence. In doing so, we unveil the tight connection between conditional models and zeroprobability events. 1.
Natural extension as a limit of regular extensions
"... This paper is devoted to the extension of conditional assessments that satisfy some consistency criteria, such as weak or strong coherence, to further domains. In particular, we characterise the natural extension of a number of conditional lower previsions on finite spaces, by showing that it can be ..."
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Cited by 2 (2 self)
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This paper is devoted to the extension of conditional assessments that satisfy some consistency criteria, such as weak or strong coherence, to further domains. In particular, we characterise the natural extension of a number of conditional lower previsions on finite spaces, by showing that it can be calculated as the limit of a sequence of conditional lower previsions defined by regular extension. Our results are valid for conditional lower previsions with nonlinear domains, and allow us to give an equivalent formulation of the notion of coherence in terms of credal sets.
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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Cited by 2 (2 self)
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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 1 (1 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.
EGL2U: Tractable Inference on Large Scale Credal Networks
, 2008
"... Credal networks [1, 2] generalize Bayesian networks [3] by associating with variables (closed convex) sets of conditional probability mass functions, i.e., credal sets 1, in place of precise conditional probability distributions. Credal networks are models of imprecise probabilities [4], which allow ..."
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Credal networks [1, 2] generalize Bayesian networks [3] by associating with variables (closed convex) sets of conditional probability mass functions, i.e., credal sets 1, in place of precise conditional probability distributions. Credal networks are models of imprecise probabilities [4], which allow the capturing