A Course on Borel Sets

Srivastava, S. M.

doi:10.1007/b98956

Cited by 84 publications

(4 citation statements)

References 82 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A Polish space is a topological space that is separable (i. e. there exists a countable dense set), and completely metrizable (i. e. there is a complete metric on the space generating its topology). Polish and (the later introduced) standard Borel spaces are introduced and treated in detail in [Kec95, Chapter 1 & 2] and [Sri98]. In this paper, we heavily exploit the properties of Polish spaces.…”

Section: Concept Notationmentioning

confidence: 99%

Infinite Probabilistic Databases

Grohe¹,

Lindner²

2022

Logical Methods in Computer Science

View full text Add to dashboard Cite

Probabilistic databases (PDBs) model uncertainty in data in a quantitative way. In the established formal framework, probabilistic (relational) databases are finite probability spaces over relational database instances. This finiteness can clash with intuitive query behavior (Ceylan et al., KR 2016), and with application scenarios that are better modeled by continuous probability distributions (Dalvi et al., CACM 2009). We formally introduced infinite PDBs in (Grohe and Lindner, PODS 2019) with a primary focus on countably infinite spaces. However, an extension beyond countable probability spaces raises nontrivial foundational issues concerned with the measurability of events and queries and ultimately with the question whether queries have a well-defined semantics. We argue that finite point processes are an appropriate model from probability theory for dealing with general probabilistic databases. This allows us to construct suitable (uncountable) probability spaces of database instances in a systematic way. Our main technical results are measurability statements for relational algebra queries as well as aggregate queries and Datalog queries.

show abstract

Section: Concept Notationmentioning

confidence: 99%

Infinite Probabilistic Databases

Grohe¹,

Lindner²

2022

Logical Methods in Computer Science

View full text Add to dashboard Cite

show abstract

“…In the following, we concentrate on the background from measure theory. The definitions and statements are based upon [59] and Chapter 1 of [42].…”

Section: Topology and Measure Theorymentioning

confidence: 99%

“…It is a well known fact that there are Borel sets B ⊆ R 2 such that the projection proj 1 (B) is not a Borel set in R (see [59,Theorem 4.1.5]). For such sets B, the query Q B is not measurable.…”

Section: The Possible Worlds Semantics Of Queries and Viewsmentioning

confidence: 99%

“…The straightforward approach would be to take a counting event C(F, n) in the output measurable space and to decompose F into its "left and right parts" F 1 ⊆ facts S (R 1 ) and F 2 ⊆ facts S (R 2 ) such that the instances from the preimage of the query are exactly those with # D (F 1 ) = n 1 and # D (F 2 ) = n 2 such that n 1 • n 2 = n, similar to the setting of Lemma 13. This approach does not settle the case since the sets F 1 and F 2 need not be measurable in general (see [59,Theorem 4.1.5]; we used the same argument in Example 8) which in particular violates the second precondition of Lemma 13.…”

Section: Lemma 15 Reordering Attributes Within the Type Of A Relation...mentioning

confidence: 99%

See 1 more Smart Citation

Infinite Probabilistic Databases

Grohe¹,

Lindner²

2019

Preprint

View full text Add to dashboard Cite

Probabilistic databases (PDBs) are used to model uncertainty in data in a quantitative way. In the standard formal framework, PDBs are finite probability spaces over relational database instances. It has been argued convincingly that this is not compatible with an open world semantics (Ceylan et al., KR 2016) and with application scenarios that are modeled by continuous probability distributions (Dalvi et al., CACM 2009).We recently introduced a model of PDBs as infinite probability spaces that addresses these issues (Grohe and Lindner, PODS 2019). While that work was mainly concerned with countably infinite probability spaces, our focus here is on uncountable spaces. Such an extension is necessary to model typical continuous probability distributions that appear in many applications. However, an extension beyond countable probability spaces raises nontrivial foundational issues concerned with the measurability of events and queries and ultimately with the question whether queries have a well-defined semantics.It turns out that so-called finite point processes are the appropriate model from probability theory for dealing with probabilistic databases. This model allows us to construct suitable (uncountable) probability spaces of database instances in a systematic way. Our main technical results are measurability statements for relational algebra queries as well as aggregate queries and datalog queries.

show abstract

Measurability and Products of Random Sets

Terán

2022

Building Bridges Between Soft and Statistical Methodologies for Data Science

View full text Add to dashboard Cite

A Course on Borel Sets

Cited by 84 publications

References 82 publications

Infinite Probabilistic Databases

Infinite Probabilistic Databases

Infinite Probabilistic Databases

Measurability and Products of Random Sets

Contact Info

Product

Resources

About