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Accept–Reject Methods

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Independent Random Sampling Methods

Part of the book series: Statistics and Computing ((SCO))

Abstract

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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Notes

  1. 1.

    Otherwise, if \(c_{\pi }=\int _{\mathcal {D}} \pi (x)dx \neq 1\), the acceptance rate is given by the more general expression \({\hat a}=\frac {c}{Lc_\pi }\).

  2. 2.

    However, there exist some variations in the literature [34], [41, Chap. 4] that may extend the applicability of this technique.

  3. 3.

    Indeed, as discussed in Sect. 3.2.4, the simplest scenario to use the RS algorithm occurs when the target density is bounded with bounded support. In this case, the proposal π(x) can be a uniform pdf and other RS schemes such as the band rejection method (Sect. 3.4) can be applied.

  4. 4.

    We assume normalized functions for simplicity of presentation. Unnormalized, integrable functions can be handled as well.

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Authors and Affiliations

  1. Department of Signal Theory and Communications, Carlos III University of Madrid, Madrid, Spain

    Luca Martino & Joaquín Míguez

  2. Department of Signal Theory and Communications, Technical University of Madrid, Madrid, Spain

    David Luengo

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  1. Luca Martino
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  2. David Luengo
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  3. Joaquín Míguez
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Martino, L., Luengo, D., Míguez, J. (2018). Accept–Reject Methods. In: Independent Random Sampling Methods. Statistics and Computing. Springer, Cham. https://doi.org/10.1007/978-3-319-72634-2_3

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