QSindependent

QSindependent is a term used in probability theory and statistics to describe a form of independence that holds under model uncertainty. Broadly, it denotes a relationship in which a collection of random elements behaves independently across a specified family of probability measures, rather than under a single fixed measure. In practice, QSindependence is often formulated within frameworks such as robust probability or sublinear (or nonlinear) expectation theory, where one analyzes how joint distributions factorize across all measures in a given set.

Informally, a set of random variables is QSindependent if the joint law factors into the product of the marginals for every admissible measure in the modeled family. This generalizes classical independence, which requires factorization under one fixed probability measure, by requiring a uniform factorization across multiple plausible models. The notion is closely related to concepts like quasi-sure analysis, polar sets, and independence under nonlinear expectations, and it can be stronger or weaker than standard independence depending on the context and the chosen measure family.

Key properties of QSindependence include its compatibility with classical independence (QSindependence implies ordinary independence in certain settings) and its sensitivity to the specified set of measures. It is typically preserved under specific transformations that do not alter the underlying model uncertainty, but can be delicate to verify in practice.

Applications of QSindependence arise in risk management, financial mathematics with model uncertainty, and machine learning scenarios where distribution drift or ambiguity between models is a concern. Critics note that the concept can be abstract and challenging to verify, with results heavily depending on how the measure family is defined.

See also: independence, quasi-sure analysis, sublinear expectation, robust statistics.