# A proof of rejection sampling correctness

Let $$f\left(x\right)$$ be a density and $$\pi\left(x\right)$$ be a function satisfying $$0\leq\pi\left(x\right)\leq1$$. In other words, $$\pi\left(x\right)$$ is a probability for every $$x$$. Then $$g\left(x\right)\propto f\left(x\right)\pi\left(x\right)$$ is density, since $$\rho=\int f\left(x\right)\pi\left(x\right)dx<1$$. This is an example of a , a class of models introduced by Rao (1965). Since $$p\left(x\right)$$ is a probability, we can call this a . This note views rejection sampling (Neumann 1951) as sampling from a particular sort of probability weighted density.

#### Jonas Moss

Meta-contrarian statistician who wants to save science.

Ph.d. student in statistics, Universitas Osloensis

Oslo in Norway