Statistics and Computing

, Volume 21, Issue 4, pp 633–647 | Cite as

A generalization of the adaptive rejection sampling algorithm

Article

Abstract

Rejection sampling is a well-known method to generate random samples from arbitrary target probability distributions. It demands the design of a suitable proposal probability density function (pdf) from which candidate samples can be drawn. These samples are either accepted or rejected depending on a test involving the ratio of the target and proposal densities. The adaptive rejection sampling method is an efficient algorithm to sample from a log-concave target density, that attains high acceptance rates by improving the proposal density whenever a sample is rejected. In this paper we introduce a generalized adaptive rejection sampling procedure that can be applied with a broad class of target probability distributions, possibly non-log-concave and exhibiting multiple modes. The proposed technique yields a sequence of proposal densities that converge toward the target pdf, thus achieving very high acceptance rates. We provide a simple numerical example to illustrate the basic use of the proposed technique, together with a more elaborate positioning application using real data.

Keywords

Rejection sampling Adaptive rejection sampling Gibbs sampling Monte Carlo integration Sensor networks Target localization 

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References

  1. Ali, A.M., Yao, K., Collier, T.C., Taylor, E., Blumstein, D., Girod, L.: An empirical study of collaborative acoustic source localization. In: Proceedings Information Processing in Sensor Networks, IPSN07, Boston (2007) Google Scholar
  2. Devroye, L.: Non-uniform Random Variate Generation. Springer, Berlin (1986) MATHGoogle Scholar
  3. Evans, M., Swartz, T.: Random variate generation using concavity properties of transformed densities. J. Comput. Graph. Stat. 7(4), 514–528 (1998) CrossRefGoogle Scholar
  4. Gilks, W.R.: Derivative-free adaptive rejection sampling for Gibbs sampling. Bayesian Stat. 4, 641–649 (1992) Google Scholar
  5. Gilks, W.R., Wild, P.: Adaptive rejection sampling for Gibbs sampling. J. R. Stat. Soc., Ser. C Appl. Stat. 41(2), 337–348 (1992) MATHGoogle Scholar
  6. Gilks, W.R., Robert, N.G.O., George, E.I.: Adaptive direction sampling. J. R. Stat. Soc., Ser. D Stat. 43(1), 179–189 (1994) Google Scholar
  7. Gilks, W.R., Best, N.G., Tan, K.K.C.: Adaptive rejection metropolis sampling within Gibbs sampling. J. R. Stat. Soc., Ser. C Appl. Stat. 44(4), 455–472 (1995) MATHGoogle Scholar
  8. Görür, D., Teh, Y.W.: Concave convex adaptive rejection sampling. Technical Report, University College, London (2009) Google Scholar
  9. Hörmann, W.: A rejection technique for sampling from t-concave distributions. ACM Trans. Math. Softw. 21(2), 182–193 (1995) MATHCrossRefGoogle Scholar
  10. Kotecha, J.H., Djurić, P.M.: Gibbs sampling approach for generation of truncated multivariate Gaussian random variables. In: Proceedings of the IEEE ICASSP, vol. 3, pp. 1757–1760 (1999) Google Scholar
  11. Künsch, H.R.: Recursive Monte Carlo filters: Algorithms and theoretical bounds. Ann. Stat. 33(5), 1983–2021 (2005) MATHCrossRefGoogle Scholar
  12. Mayo, P., Rodenas, F., Verdú, G.: Comparing methods to denoise mammographic images. In: Proceedings of the 26th IEEE EMBS, vol. 1, pp. 247–250 (2004) Google Scholar
  13. Patwari, N., Hero, A.O., Perkins, M., Correal, N.S., O’Dea, R.J.: Relative location estimation in wireless sensor networks. IEEE Trans. Signal. Process. 51(5), 2137–2148 (2003) CrossRefGoogle Scholar
  14. Rappaport, T.S.: Wireless Communications: Principles and Practice, 2nd edn. Prentice-Hall, Upper Saddle River (2001) Google Scholar
  15. Reiss, R., Thomas, M.: Statistical Analysis of Extreme Values: With Applications to Insurance, Finance, Hydrology and Other Fields. Springer, Berlin (2007) MATHGoogle Scholar
  16. Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, Berlin (2004) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Signal Theory and CommunicationsUniversidad Carlos III de MadridLeganésSpain

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