Probabilistic Logic in a Coherent Setting

Coletti, Giulianella; Scozzafava, Romano

doi:10.1007/978-94-010-0474-9

Cited by 235 publications

(228 citation statements)

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“…Then the independent natural extension P A 1 1 ⊗ P 2 is coherent with P {1,2} (·|X 1 ), so we infer from Eq. (4) that P A 1 1 ⊗ P 2 ≥ (P A 1 1 ⊗ P 2 )(P {1,2} (·|X 1 )) = P A 1 1 (P {1,2} (·|X 1 )), where the equality follows from Corollary 15.…”

Section: It Follows From Lemma 35(iii) Lemma 34 and Fmentioning

confidence: 93%

“…The first equality then follows by letting h 1 := g and h 2 := g(·, x I ). It is clear from SC1-SC3 that P O∪I (·|X I ) is separately coherent if and only if for all x I ∈ X I , P O∪I (·|x I ) is a coherent lower prevision on L (X O ) and moreover Condition (1) holds [this second condition turns out to be equivalent to requiring that P O∪I ({x I }|x I ) = 1 for every x I ∈ X I ].…”

Section: 3mentioning

confidence: 99%

“…Let P A 1 1 be the vacuous lower prevision on L (X 1 ) relative to the nonempty set A 1 ⊆ X 1 , and let P 2 be any coherent lower prevision on L (X 2 ). For all gambles f on X 1 × X 2 :…”

Section: 5mentioning

confidence: 99%

“…1 But it turns out that we lose this formal equivalence as soon as we consider probabilities that may be imprecisely specified, meaning that the available information is conveniently expressed through sets of probabilities (sets of mass functions). In this case the two routes diverge also mathematically, as we shall see further on.…”

Section: Introductionmentioning

confidence: 99%

“…This point is not without importance, as it shows that buying and selling prices for gambles are actually a more primitive and fundamental notion in a theory of personal probability. 1 There may be subtleties, however, related to events of probability zero. See Refs.…”

Section: Introductionmentioning

confidence: 99%

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Independent Natural Extension

Cooman

Miranda

Zaffalon

2010

Computational Intelligence for Knowledge-Based Systems Design

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ABSTRACT. There is no unique extension of the standard notion of probabilistic independence to the case where probabilities are indeterminate or imprecisely specified. Epistemic independence is an extension that formalises the intuitive idea of mutual irrelevance between different sources of information. This gives epistemic independence very wide scope as well as appeal: this interpretation of independence is often taken as natural also in preciseprobabilistic contexts. Nevertheless, epistemic independence has received little attention so far. This paper develops the foundations of this notion for variables assuming values in finite spaces. We define (epistemically) independent products of marginals (or possibly conditionals) and show that there always is a unique least-committal such independent product, which we call the independent natural extension. We supply an explicit formula for it, and study some of its properties, such as associativity, marginalisation and external additivity, which are basic tools to work with the independent natural extension. Additionally, we consider a number of ways in which the standard factorisation formula for independence can be generalised to an imprecise-probabilistic context. We show, under some mild conditions, that when the focus is on least-committal models, using the independent natural extension is equivalent to imposing a so-called strong factorisation property. This is an important outcome for applications as it gives a simple tool to make sure that inferences are consistent with epistemic independence judgements. We discuss the potential of our results for applications in Artificial Intelligence by recalling recent work by some of us, where the independent natural extension was applied to graphical models. It has allowed, for the first time, the development of an exact linear-time algorithm for the imprecise probability updating of credal trees.

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Section: It Follows From Lemma 35(iii) Lemma 34 and Fmentioning

confidence: 93%

Section: 3mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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