Densityto

Densityto is a term used in theoretical statistics and applied mathematics to refer to a class of transformations that relate densities across spaces. Broadly, a densityto transformation maps a density p on a domain X to a density q on a domain Y via a measurable mapping f: X → Y, preserving total mass under appropriate regularity conditions. In intuitive terms, densityto describes how probability mass is redistributed when the underlying variables are transformed.

In discrete settings, if X is finite and f maps X to Y, q(y) = Σ_{x: f(x)=y} p(x). In continuous settings, for a smooth f with Jacobian J, the density on Y is q(y) = ∑_{x: f(x)=y} p(x)/|det J_f(x)|, described similarly as the pushforward of the density p by f. The densityto framework emphasizes the relationship between densities under transformation, rather than the original random variable.

Origins and usage: The term densityto has appeared in recent lecture notes and research articles to discuss density propagation, sometimes interchangeably with the pushforward of measures. Different authors emphasize different conditions, leading to variations in definitions; there is no single canonical formulation.

Applications: Densityto concepts appear in probabilistic modeling, statistical inference, and image or signal processing where one needs to propagate or transform density information through a model, for example in forward models, data assimilation, or density-based clustering.

See also: probability density function, pushforward measure, change of variables formula, transformation of random variables, Bayesian inference.