DiferencimP

DiferencimP is a statistical measure used to quantify differences between two datasets or probability distributions. The term combines "difference" with the initial P, signaling its basis in a penalized comparison framework. It is discussed mainly in data analysis and machine learning contexts.

Definition and framework: DiferencimP(D1,D2) is defined as the supremum over a class F of functions f of [∫ f d(D1−D2) − P(f)], where P is a penalty term that discourages overly complex discriminators. F is typically restricted by smoothness or Lipschitz conditions.

Relation to other metrics: Depending on the choice of F and P, DiferencimP generalizes several integral probability metrics. With a convex penalty that encourages smooth discriminators, it resembles maximum mean discrepancy; with a Lipschitz constraint it connects to Wasserstein-type distances.

Computation: In practice, DiferencimP is estimated from finite samples by solving a constrained optimization problem to find a near-optimal function f in F. Kernel methods, neural network approximations, or convex relaxations are commonly used.

Applications: The measure is used for comparing datasets, anomaly detection, change-point analysis, domain adaptation, model criticism, and evaluation of generative models where controlled discrimination is desirable.

History and variants: The concept emerged in theoretical discussions in the 2010s and 2020s as part of a broader class of penalized IPMs. Variants differ in the choice of F and the form of the penalty P, yielding different sensitivity properties.