Sufficiency

Sufficiency is the quality of having enough for a particular purpose. The term appears in philosophy, logic, statistics, and everyday language to describe conditions or information that guarantee a result. It is often contrasted with necessity; a condition can be sufficient without being necessary, and vice versa.

In logic and philosophy, A is a sufficient condition for B if A implies B. For example, being a square is a sufficient condition for being a rectangle: every square is a rectangle, but not every rectangle is a square.

In statistics, sufficiency has a precise meaning. A statistic T(X) is sufficient for a parameter θ if the conditional distribution of the data X given T(X) does not depend on θ. A standard criterion is the factorization theorem: the joint density can be written as f(x; θ) = g(T(x); θ) h(x). This means T(x) captures all information about θ contained in the sample. Examples include the sample mean for the normal mean with known variance, and natural sufficient statistics in exponential families. Concepts such as minimal sufficiency and complete sufficiency refine data reduction and estimation.

In practice, sufficiency is context-dependent. It identifies what is enough to determine outcomes, but it does not by itself ensure optimal decisions or unique conclusions. It is often used alongside other criteria in statistical analysis and reasoning.