A logical characterization of coherence for imprecise probabilities

Fedel, Martina; Hosni, Hykel; Montagna, Franco

doi:10.1016/j.ijar.2011.06.004

Cited by 11 publications

(6 citation statements)

References 27 publications

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“…This generalization encompasses our notion of paraconsistent probability based on the logic Ci as well as paraconsistent probability theories based on several other LFIs. More recently, it has been shown in [42] that the Dutch book argument can be further extended to the domain of MV-algebras, providing a logical characterization of coherence for imprecise probability.…”

Section: Summary Comments and Conclusionmentioning

confidence: 99%

Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

Bueno-Soler

Carnielli

2016

Entropy

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This paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs). We show that LFIs encode very naturally an extension of the notion of probability able to express sophisticated probabilistic reasoning under contradictions employing appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes' theorem for conditionalization. We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Some critical historical and conceptual points about probability theory are also reviewed.

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Section: Summary Comments and Conclusionmentioning

confidence: 99%

Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

Bueno-Soler

Carnielli

2016

Entropy

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“…So, a+N M ∈ A * , and it suffices to define P * (a) = M ·S * ( a+N M )−N , and U * (a) = M ·U * 0 ( a+N M )−N . Note that in [5] it is shown that the definition does not depend on the choice of the integers M and N such that a+N M ∈ A. (3) Let U * 0 be an upper hyperstate on A * and U * be the unique upper hyperprevision on G * extending U * 0 .…”

Section: Fuzzy Imprecise Probabilities Over the Hyperrealsmentioning

confidence: 99%

Stable Non-standard Imprecise Probabilities

Hosni

Montagna²

2014

Information Processing and Management of Uncertainty in Knowledge-Based Systems

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Abstract. Stability arises as the consistency criterion in a betting interpretation for hyperreal imprecise previsions, that is imprecise previsions (and probabilities) which may take infinitesimal values. The purpose of this work is to extend the notion of stable coherence introduced in [8] to conditional hyperreal imprecise probabilities. Our investigation extends the de Finetti-Walley operational characterisation of (imprecise) prevision to conditioning on events which are considered "practically impossible" but not "logically impossible".

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“…Publishing books with zero expectation is necessary and sufficient to protect the bookmaker from being forced into sure loss by rational gamblers, possibly through Swapping. Fedel et al (2011) investigate the relaxation of this first-order uncertainty modelling feature by considering a betting scenario in which bookmakers are motivated by making profit in a market-like environment. This clearly imposes the relaxation of Swapping so that gamblers can no longer choose the sign of the stake for their bets.…”

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confidence: 99%

“…38 How can the Choice Norm be instantiated for this (second-order uncertainty) for-profit betting problem? A simultaneous extension of classical Bayesianism to imprecise and fuzzy probabilities is carried out in (Fedel et al, 2011) by constructing the analytic framework of imprecise probabilities on top of a many-valued algebraic semantics. For present purposes I will limit myself to an informal discussion of how the notion of admissibility is arrived at and refer the interested reader to the original paper for precise mathematical details and for further motivation concerning the extension to fuzzy events.…”

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confidence: 99%

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Towards a Bayesian Theory of Second-Order Uncertainty: Lessons from Non-Standard Logics

Hosni

2013

David Makinson on Classical Methods for Non-Classical Problems

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Abstract. Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are illsuited to represent a number of interesting second-order uncertainty features, especially "ignorance" and "ambiguity". This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes vs anti-Bayes debate which is in many ways analogous to what the classical vs non-classical debate used to be in logic. This paper attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisation of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for extending the expressive power of standard Bayesian norms of rationality to secondorder uncertainty in a way which is both formally and foundationally conservative.

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A logical characterization of coherence for imprecise probabilities

Cited by 11 publications

References 27 publications

Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

Stable Non-standard Imprecise Probabilities

Towards a Bayesian Theory of Second-Order Uncertainty: Lessons from Non-Standard Logics

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