Distributiondivision

Distributiondivision is an informal term used to describe operations that form a new distribution by dividing two input distributions. In probability theory and statistics, the most central concept related to this idea is the likelihood ratio, the pointwise ratio of two probability density functions (continuous case) or probability mass functions (discrete case).

Definition: Let f and g be PDFs on a domain X with g(x) > 0 for all x in X. The ratio r(x) = f(x)/g(x) is the likelihood ratio. It is not itself a probability distribution unless normalized. To obtain a new distribution from the ratio, set h(x) ∝ f(x)/g(x) and normalize: h(x) = (f(x)/g(x)) / ∫ (f(u)/g(u)) du, provided the integral is finite.

Connections: This ratio is the Radon-Nikodym derivative of the measure with density f with respect to the measure with density g, when f is absolutely continuous with respect to g. In statistics, the likelihood ratio is central to hypothesis testing and model comparison; in sampling, division by g(x) yields importance sampling weights.

Examples: If f and g are normal distributions with the same variance, the ratio f(x)/g(x) simplifies to an exponential function of x, reflecting the relative evidence for two models across the support.

Limitations: The ratio can be undefined where g=0; the unnormalized ratio is not a probability distribution and may not be integrable without normalization; interpretation depends on the statistical context.

See also: likelihood ratio, Radon-Nikodym derivative, importance sampling, change of measure.