Sets

Schechter, Eric

doi:10.1016/b978-012622760-4/50001-7

Cited by 1 publication

(5 citation statements)

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“…Certainly, every subset

is compatible if and only if every finite subset of

is compatible. Hence, compatibility is a property of so-called finite character [ 25 ] (Definition 3.46). Then, Tuckey’s lemma (e.g., [ 25 ] (Theorem 6.20.AC5)) guarantees that any compatible subset of

is contained in a maximal compatible subset.…”

Section: What Is a (Pre-)dynkin System?mentioning

confidence: 99%

“…Hence, compatibility is a property of so-called finite character [ 25 ] (Definition 3.46). Then, Tuckey’s lemma (e.g., [ 25 ] (Theorem 6.20.AC5)) guarantees that any compatible subset of

is contained in a maximal compatible subset. Since every element

is in at least one compatible subset, e.g.…”

Section: What Is a (Pre-)dynkin System?mentioning

confidence: 99%

“…In addition, the two functions form a Galois connection. In fact, every Galois connection defines closure operators, i.e., extensive, monotone and idempotent maps [ 25 ] (Definition 4.5.a). The closure operators are defined as the sequential application of the credal set function and the dual credal set function to subsets of

.…”

Section: The Credal Set and Its Relation To Pre-dynkin System Structurementioning

confidence: 99%

“…Let

be a countably additive probability on

. For each σ-block

we assume that

is inner regular, i.e., if

[ 25 ] (p. 808). The probability

is σ-extendable if and only if

for all finite families of measurable simple gambles

.…”

Section: Appendix A1 Compatibility Structurementioning

confidence: 99%

“…For this reason, we use, with minor abuse of notation, the same notation for bounded, signed, finitely additive measures and for continuous linear functionals in the dual space of

, i.e., we write

. The dual space

gets equipped with the weak

topology, i.e., the weakest topology which makes all evaluation functionals of the form

such that

for some

continuous (for more details see, e.g., [ 25 ] (p. 758)). With

, we denote the convex, weak

-closed subset of finitely additive probability measures.…”

Section: Introductionmentioning

confidence: 99%

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Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces

Derr,

Williamson

2023

Entropy

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In the literature on imprecise probability, little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems 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. Here, (pre-)Dynkin systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin systems turns out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin systems. Again, coherence and extendability are equivalent. A related but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities.

show abstract

“…Certainly, every subset

is compatible if and only if every finite subset of

is contained in a maximal compatible subset.…”

Section: What Is a (Pre-)dynkin System?mentioning

confidence: 99%

“…Hence, compatibility is a property of so-called finite character [ 25 ] (Definition 3.46). Then, Tuckey’s lemma (e.g., [ 25 ] (Theorem 6.20.AC5)) guarantees that any compatible subset of

is contained in a maximal compatible subset. Since every element

is in at least one compatible subset, e.g.…”

Section: What Is a (Pre-)dynkin System?mentioning

confidence: 99%

.…”

Section: The Credal Set and Its Relation To Pre-dynkin System Structurementioning

confidence: 99%

“…Let

be a countably additive probability on

. For each σ-block

we assume that

is inner regular, i.e., if

[ 25 ] (p. 808). The probability

is σ-extendable if and only if

for all finite families of measurable simple gambles

.…”

Section: Appendix A1 Compatibility Structurementioning

confidence: 99%

“…For this reason, we use, with minor abuse of notation, the same notation for bounded, signed, finitely additive measures and for continuous linear functionals in the dual space of

, i.e., we write

. The dual space

gets equipped with the weak

topology, i.e., the weakest topology which makes all evaluation functionals of the form

such that

for some

continuous (for more details see, e.g., [ 25 ] (p. 758)). With

, we denote the convex, weak

-closed subset of finitely additive probability measures.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces

Derr,

Williamson

2023

Entropy

View full text Add to dashboard Cite

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Sets

Cited by 1 publication

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Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces

Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces

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