Distributionand

Distributionand is a term used in some theoretical and educational discussions of probability to denote a constrained operator that combines two distributions under a logical conjunction. It is not an established standard in mainstream probability, but it appears in tutorials and exploratory writings as a compact way to discuss constrained joint distributions.

Definition and notation

Let X have distribution D1 with density f1(x) (or mass function p1(x)) on a space X, and Y have distribution D2 with density f2(y) on a space Y. Let g: X × Y → {true, false} be a predicate that imposes a constraint. The distributionand of D1 and D2 under g, denoted D1 distributionand D2, is the joint distribution on X × Y with density (or mass) proportional to f1(x) f2(y) whenever g(x,y) is true, and zero otherwise. Formally, f_AND(x,y) ∝ f1(x) f2(y) 1{g(x,y)}, with normalization Z = ∑_{x,y} f1(x) f2(y) 1{g(x,y)} or the corresponding integral. If using discrete distributions, sums replace integrals.

Relation to existing concepts

Distributionand is a formalization of forming a joint distribution under a constraint, akin to conditioning or metakind of joint construction. If g(x,y) is always true, distributionand reduces to the ordinary product distribution (the independent joint). If g imposes a restriction, the marginals of X and Y under the constrained joint generally differ from D1 and D2.

Applications and limitations

In probabilistic programming and Bayesian inference, distributionand provides a convenient way to model sampling under relational predicates or constraints. It emphasizes the normalization constant and the dependence introduced by the predicate. Its use varies by author and is not universally standardized, serving mainly as a conceptual tool rather than a canonical construct.

See also: joint distribution, conditional distribution, constrained sampling, copulas.