A survey of concepts of independence for imprecise probabilities

Couso, Inés; Moral, Serafı́n; Walley, Peter

doi:10.1017/s1357530900000156

Cited by 142 publications

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“…[2]. This illustrates that a behavioural theory of probability and the notion of epistemic independence fit nicely together.…”

Section: Introductionmentioning

confidence: 70%

“…Because we are dealing with linear previsions, coherence is equivalent to the condition in Equation (2). Assume ex absurdo that there are f O,I ∈ L (X O∪I ) for all disjoint subsets I and O of N, and δ > 0 such that…”

Section: This Clearly Implies Thatmentioning

confidence: 99%

“…We conclude with a proof for the third statement. Assume that P 2 is the vacuous lower prevision P A 2 2 with respect to A 2 , and let P be any coherent lower prevision on L (X {1,2} ) with marginals P A 1 1 and P A 2 2 . Because it is coherent, P satisfies…”

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

confidence: 99%

“…This is exemplified by the existence of the different notions of strong and epistemic independence, respectively. 2 Of these two, strong independence has been most thoroughly investigated in the literature. Studies of epistemic independence are confined to a relatively small number of papers [6,10,26,29] inspired by Peter Walley's [30, Section 9.3] seminal ideas.…”

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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“…[2]. This illustrates that a behavioural theory of probability and the notion of epistemic independence fit nicely together.…”

Section: Introductionmentioning

confidence: 70%

Section: This Clearly Implies Thatmentioning

confidence: 99%

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

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Independent Natural Extension

Cooman

Miranda

Zaffalon

2010

Computational Intelligence for Knowledge-Based Systems Design

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“…In general, determining whether a Bayes net satisfies CMC and Minimality requires an understanding of what it means for two random variables to be independent of one another. In the precise context, such independence has only one mathematical definition, but in an imprecise context, multiple independence concepts abound (see [7,8]). Most saliently, imprecise probability distributions over different variables may satisfy so-called strong independence, or the weaker notion of epistemic independence.…”

Section: Introductionmentioning

confidence: 99%

Imprecise Bayesian Networks as Causal Models

Kinney

2018

Information

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This article considers the extent to which Bayesian networks with imprecise probabilities, which are used in statistics and computer science for predictive purposes, can be used to represent causal structure. It is argued that the adequacy conditions for causal representation in the precise context-the Causal Markov Condition and Minimality-do not readily translate into the imprecise context. Crucial to this argument is the fact that the independence relation between random variables can be understood in several different ways when the joint probability distribution over those variables is imprecise, none of which provides a compelling basis for the causal interpretation of imprecise Bayes nets. I conclude that there are serious limits to the use of imprecise Bayesian networks to represent causal structure.

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A survey of concepts of independence for imprecise probabilities

Cited by 142 publications

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Independent Natural Extension

Independent Natural Extension

Imprecise Bayesian Networks as Causal Models

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