Extending States on Finite Concrete Logics

Simone, Anna De; Navara, Mirko; Pták, Pavel

doi:10.1007/s10773-006-9298-6

Cited by 11 publications

(13 citation statements)

References 11 publications

(6 reference statements)

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“…Extendability of probabilities on (pre-)Dynkin-systems has already been part of discussions in quantum probability since 1969 [Gudder, 1969] up to more current times [De Simone and Pták, 2010]. Several necessary and/or sufficient conditions on the structure of D and/or the values of µ are known [Gudder, 1984, De Simone et al, 2007, De Simone and Pták, 2010. We present here a sufficient and necessary condition discovered by Horn and Tarski [1948].…”

Section: Extendabilitymentioning

confidence: 88%

“…The definition is non-vacuous [Gudder, 1984, De Simone et al, 2007. For instance, a probability measure on a pre-Dynkin-System is not generally extendable to a measure on the generated algebra (e.g.…”

Section: Extendabilitymentioning

confidence: 99%

“…pre-Dynkin-system [Schurz and Leitgeb, 2008] additive-class [Rao and Rao, 1983, p. 2] concrete logic [Ovchinnikov, 1999, De Simone et al, 2007 partial field [Godowski, 1981] quantum-mechanical algebra [Suppes, 1966] semi-algebra [Khrennikov, 2009a, p. 13] set-representable orthomodular poset (SROMP) [Pták, 1998] Table 2: A short summary of names for pre-Dynkin-systems found in literature. Dynkin-system [Jun et al, 2011] d-system [Williams, 1991, p. 193] λ-class [Chow and Teicher, 1988, p. 7] quantum-mechanical σ-algebra [Suppes, 1966] σ-class [Gudder, 1984] Table 3: A short summary of names for Dynkin-systems found in literature.…”

Section: B Names Of Pre-dynkin-systems and Dynkin-systemsmentioning

confidence: 99%

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The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems

Derr¹,

Williamson²

2023

Preprint

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In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkinsystems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum physics are equivalent to coherence in the sense of Walley [Walley, 1991, p. 84]. Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities. When posing problems in probability calculus,it should be required to indicate for which events the probabilities are assumed to exist.

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Section: Extendabilitymentioning

confidence: 88%

Section: Extendabilitymentioning

confidence: 99%

Section: B Names Of Pre-dynkin-systems and Dynkin-systemsmentioning

confidence: 99%

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The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems

Derr¹,

Williamson²

2023

Preprint

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“…The symmetric logics and certain states on them have been studied in . The notion of the

Δ

‐subadditive state seems to first occur in . Some results about

Δ

‐subadditive states can be found in , , , , .…”

mentioning

confidence: 99%

“…Utilizing the property that

\tilde{s} (A Δ B) \leq \tilde{s} (A) + \tilde{s} (B)

, it is not difficult to show that if we set, for distinct

x, u, v \in Ω_{2 n}

p (x) = \frac{1}{2} (\overset{s}{true} ({x, u}) + \overset{s}{true} ({x, v}) - \overset{s}{true} ({u, v})),

we find out that this value does not depend on the choice of

u, v

, and, moreover, if we define a state w on

(Ω_{2 n}, exp Ω_{2 n})

by putting

w (A) = \sum_{a \in A} p (a)

, we obtain an extension of

\overset{s}{true}

over

(Ω_{2 n}, exp Ω_{2 n})

. (For details see ; the paper provides another method for proving this result.) Using the isomorphism between

(Ω, {scriptD}_{1})

and

(Ω_{2 n}, {scriptE}_{2 n})

, we obtain an extension of s from

(Ω, {scriptD}_{1})

to a Boolean algebra of subsets of Ω corresponding to

(Ω_{2 n}, exp Ω_{2 n})

.…”

mentioning

confidence: 99%

States on systems of sets that are closed under symmetric difference

2015

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We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set-representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non-Boolean situations. On the other hand, a difference-closed system can be constructed such that even two-valued states do not allowfor extensions. Finally,we consider these questions in a σ-complete setup and find a large class of such systems with rather interesting state properties

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