dc.contributor.author
Klebanov, Ilja
dc.date.accessioned
2024-10-07T09:37:26Z
dc.date.available
2024-10-07T09:37:26Z
dc.identifier.uri
https://refubium.fu-berlin.de/handle/fub188/44196
dc.identifier.uri
http://dx.doi.org/10.17169/refubium-43906
dc.description.abstract
Jensen’s inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function𝑓∶ 𝐾 → ℝ defined on a convex domain 𝐾 ⊆ ℝ𝑑 and any random variable 𝑋 taking values in 𝐾, 𝔼[𝑓(𝑋)] ⩾ 𝑓(𝔼[𝑋]). In this paper, sharp upper and lower bounds on 𝔼[𝑓(𝑋)], termed ‘graph convex hull bounds’, are derived for arbitrary functions 𝑓 on arbitrary domains 𝐾, thereby extensively generalizing Jensen’s inequality. The derivation of these bounds necessitates the investigation of the convex hull of the graph of 𝑓, which can be challenging for complex functions. On the other hand, once these inequalities are established, they hold, just like Jensen’s inequality, for any 𝐾-valued random variable 𝑋. Therefore, these bounds are of particular interest in cases where 𝑓 is relatively simple and 𝑋 is complicated or unknown. Both finite- and infinite-dimensional domains and codomains of 𝑓 are covered as well as analogous bounds for conditional expectations and Markov operators.
en
dc.format.extent
14 Seiten
dc.rights.uri
https://creativecommons.org/licenses/by-nc-nd/4.0/
dc.subject
Jensen's inequality
en
dc.subject
arbitrary functions
en
dc.subject
generalizing
en
dc.subject.ddc
500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik
dc.title
Graph convex hull bounds as generalized Jensen inequalities
dc.type
Wissenschaftlicher Artikel
dcterms.bibliographicCitation.doi
10.1112/blms.13116
dcterms.bibliographicCitation.journaltitle
Bulletin of the London Mathematical Society
dcterms.bibliographicCitation.number
10
dcterms.bibliographicCitation.pagestart
3061
dcterms.bibliographicCitation.pageend
3074
dcterms.bibliographicCitation.volume
56
dcterms.bibliographicCitation.url
https://doi.org/10.1112/blms.13116
refubium.affiliation
Mathematik und Informatik
refubium.affiliation.other
Institut für Mathematik
refubium.funding
DEAL Wiley
refubium.note.author
Die Publikation wurde aus Open Access Publikationsgeldern der Freien Universität Berlin gefördert.
refubium.resourceType.isindependentpub
no
dcterms.accessRights.openaire
open access
dcterms.isPartOf.eissn
1469-2120