The accept/reject method, also known as rejection sampling (RS), was suggested by John von Neumann in 1951. It is a classical Monte Carlo technique for universal sampling that can be used to generate samples virtually from any target density p o (x) by drawing from a simpler proposal density π(x). The sample is either accepted or rejected by an adequate test of the ratio of the two pdfs, and it can be proved that accepted samples are actually distributed according to the target distribution. Specifically, the RS algorithm can be viewed as choosing a subsequence of i.i.d. realizations from the proposal density π(x) in such a way that the elements of the subsequence have density p o (x).
In this chapter, we present the basic theory of RS as well as different variants found in the literature. Computational cost issues and the range of applications are analyzed in depth. Several combinations with other Monte Carlo techniques are also described.
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