sumtoone

Sumtoone is a term used to describe a constraint in which a collection of quantities sums to one. In mathematics and its applications, sum-to-one constraints are fundamental for representing probabilities, normalized weights, and convex combinations. A vector x = (x1, ..., xn) is sumtoone if each xi is nonnegative and the sum of all components equals 1. The set of all such vectors forms the standard (n−1)-simplex in Euclidean space.

Practically, sumtoone vectors are often obtained by normalization. If y = (y1, ..., yn) has nonnegative components, then x_i = y_i / sum_j y_j yields a sumtoone vector. The softmax function, x_i = exp(z_i) / sum_j exp(z_j), also produces a sumtoone vector from any real-valued input z. In probabilistic modeling, the Dirichlet distribution serves as a natural prior over probability vectors that lie on the simplex, ensuring the sumtoone property.

Applications of sumtoone vectors are widespread. They represent probability distributions over discrete outcomes, weights in ensemble methods and attention mechanisms, and parameters in mixture models. In economics and decision theory, they model budget shares or allocation proportions that must total one. In linear algebra and Markov processes, row-stochastic matrices have rows that are sumtoone, reflecting transition probabilities.

Related concepts include the simplex, probability distributions, normalization, and convex combinations. The core idea of sumtoone facilitates constraining and interpreting sets of components as complete, mutually exclusive parts of a whole.