Non-standard probability, coherence and conditional probability on many-valued events

Montagna, Franco; Fedel, Martina; Scianna, Giuseppe

doi:10.1016/j.ijar.2013.02.003

Cited by 9 publications

(14 citation statements)

References 18 publications

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“…As implied by [28,Theorem 4.2] this setting is immune to the limitations arising from the potential lack, in general, of faithful states of MV-algebras. that r · t < r · x.…”

Section: Definition 42 ([17])mentioning

confidence: 99%

Strict Coherence on Many-Valued Events

Flaminio¹,

Hosni²,

Montagna³

2018

J. symb. log.

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Abstract. We investigate the property of strict coherence in the setting of manyvalued logics. Our main results read as follows: (i) a map from an MV-algebra to [0,1] is strictly coherent if and only if it satisfies Carnap's regularity condition and, (ii) a [0, 1]-valued book on a finite set of many-valued events is strictly coherent if and only if it extends to a faithful state of an MV-algebra that contains them. Remarkably this latter result allows us to relax the rather demanding conditions for the Shimony-Kemeny characterisation of strict coherence put forward in the mid 1950s in this Journal. §1. Introduction and motivation. This paper contributes to the logical foundations of probability by investigating strict coherence on many-valued events. The notion of strict coherence was introduced in this Journal by Abner Shimony and John Kemeny as a logically inspired refinement of the notion of coherence used by Bruno de Finetti to ground his subjective interpretation of probability. Informally, coherence demands that a rational agent avoids the logical possibility of "sure loss" in suitably specified betting situations. Its strict counterpart, in addition, demands that each prospect of losing should be balanced by a prospect of gaining.Interest in the condition of strict coherence was prompted by Carnap's analysis of what he termed "regular" probability functions in [4] (see also [37, Chapter 10 ]). Informally those functions arise by strengthening the usual normalisation axiom of probability in the right-to-left direction. That is to say that 1 (respectively, 0) is assigned only to tautologies (respectively, contradictions). The rationale for Carnap-regular functions is that, however unlikely, possible events may happen.In [38] Shimony proved that imposing strict coherence to a map from finite boolean algebras to [0, 1] was sufficient to single out Carnap-regular functions. Shortly after, the converse was established by Kemeny in [21], a result proved independently in [24]. The Shimony-Kemeny characterisation is obtained under far more restrictive conditions than those yielding de Finetti's theorem, which 1991 Mathematics Subject Classification. Primary 03B48, Secondary 60A05, 06D35, 06F20, 52B11.

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“…As implied by [28,Theorem 4.2] this setting is immune to the limitations arising from the potential lack, in general, of faithful states of MV-algebras. that r · t < r · x.…”

Section: Definition 42 ([17])mentioning

confidence: 99%

Strict Coherence on Many-Valued Events

Flaminio¹,

Hosni²,

Montagna³

2018

J. symb. log.

Self Cite

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“…It can be proved that both (G, u) and R * f in are embeddable into Π(G, u, R * f in ) and both A and [0, 1] * are embeddable in Π(A, [0, 1] * ), see [8].…”

Section: Theoremmentioning

confidence: 99%

“…We build on [8], which can be consulted for further background on MV-algebra and related structures. 3 Specifically, we work in the framework of unital lattice ordered abelian groups, which can be represented as algebras 4 of bounded functions from a set X into a non-standard extension, R * , of R. Hence, their elements may be interpreted as bounded random variables (also called gambles in [12]).…”

Section: Introductionmentioning

confidence: 99%

“…The set of bounded random variables is closed under sum, subtraction and under the lattice operations. Since we are interested in extending the framework of [8] to imprecise conditional prevision and probability, we also need a product operation. It turns out that appropriate structure is constituted by the c-s-u-f integral domains.…”

Section: Introductionmentioning

confidence: 99%

“…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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confidence: 99%

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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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Layers of zero probability and stable coherence over Łukasiewicz events

Flaminio

Godo

2016

Soft Comput

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The notion of stable coherence has been recently introduced to characterize coherent assignments to conditional many-valued events by means of hyperreal-valued states. In a nutshell, an assignment, or book, β on a finite set of conditional events is stably coherent if there exists a coherent variant β of β such that β maps all antecedents of conditional events to a strictly positive hyperreal number, and such that β and β differ by an infinitesimal. In this paper we provide a characterization of stable coherence in terms of layers of zero probability for books on Lukasiewicz logic events.If A is a Boolean algebra and P : A → [0, 1] is a finitely additive probability on A, the conditional probability of "a given b" can be quantified as the ratio P (a | b) = P (a∧b)/P (b), whenever P (b) > 0. Obviously, the conditional probability P (· | ·) defined in this way on A × A is only a partial function and, in particular, it is not defined on all the pairs (a | b) with P (b) = 0.Krauss [10] and Nelson [22] proposed a way to overcome this problem employing nonstandard probabilities: instead of a real-valued probability P : A → [0, 1], they This paper is dedicated to the memory of Franco Montagna, an excellent mathematician and better person and friend, to whom we owe so much.

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Non-standard probability, coherence and conditional probability on many-valued events

Cited by 9 publications

References 18 publications

Strict Coherence on Many-Valued Events

Strict Coherence on Many-Valued Events

Stable Non-standard Imprecise Probabilities

Layers of zero probability and stable coherence over Łukasiewicz events

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