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Adaptive Rejection Sampling Methods

  • Luca Martino
  • David Luengo
  • Joaquín Míguez
Chapter
Part of the Statistics and Computing book series (SCO)

Abstract

This chapter is devoted to describing the class of the adaptive rejection sampling (ARS) schemes. These (theoretically) universal methods are very efficient samplers that update the proposal density whenever a generated sample is rejected in the RS test. In this way, they can produce i.i.d. samples from the target with an increasing acceptance rate that can converge to 1. As a by-product, these techniques also generate a sequence of proposal pdfs converging to the true shape of the target density. Another advantage of the ARS samplers is that, when they can be applied, the user only has to select a set of initial conditions. After the initialization, they are completely automatic, self-tuning algorithms (i.e., no parameters need to be adjusted by the user) regardless of the specific target density. However, the need to construct a suitable sequence of proposal densities restricts the practical applicability of this methodology. As a consequence, ARS schemes are often tailored to specific classes of target distributions. Indeed, the construction of the proposal is particularly hard in multidimensional spaces. Hence, ARS algorithms are usually designed only for drawing from univariate densities.

In this chapter we discuss the basic adaptive structure shared by all ARS algorithms. Then we look into the performance of the method, characterized by the acceptance probability (which increases as the proposal is adapted), and describe various extensions of the standard ARS approach which are aimed either at improving the efficiency of the method or at covering a broader class of target pdfs. Finally, we consider a hybrid method that combines the ARS and Metropolis-Hastings schemes.

References

  1. 1.
    M.S. Arulumpalam, S. Maskell, N. Gordon, T. Klapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50(2), 174–188 (2002)Google Scholar
  2. 2.
    J. Besag, P.J. Green, Spatial statistics and Bayesian computation. J. R. Stat. Soc. Ser. B 55(1), 25–37 (1993)Google Scholar
  3. 3.
    C. Botts, A modified adaptive accept-reject algorithm for univariate densities with bounded support. Technical Report, http://williams.edu/Mathematics/cbotts/Research/paper3.pdf (2010)
  4. 4.
    G.E.P. Box, G.C. Tiao, Bayesian Inference in Statistical Analysis (Wiley, New York, 1973)Google Scholar
  5. 5.
    L. Devroye, Non-uniform Random Variate Generation (Springer, New York, 1986)Google Scholar
  6. 6.
    M. Evans, T. Swartz, Random variate generation using concavity properties of transformed densities. J. Comput. Graph. Stat. 7(4), 514–528 (1998)Google Scholar
  7. 7.
    J. Geweke, Bayesian inference in econometric models using Monte Carlo integration. Econometrica 24, 1317–1399 (1989)Google Scholar
  8. 8.
    W.R. Gilks, Derivative-free adaptive rejection sampling for Gibbs sampling. Bayesian Stat. 4, 641–649 (1992)Google Scholar
  9. 9.
    W.R. Gilks, P. Wild, Adaptive rejection sampling for Gibbs sampling. Appl. Stat. 41(2), 337–348 (1992)Google Scholar
  10. 10.
    W.R. Gilks, N.G. Best, K.K.C. Tan, Adaptive rejection Metropolis sampling within Gibbs sampling. Appl. Stat. 44(4), 455–472 (1995)Google Scholar
  11. 11.
    D. Gorur, Y.W. Teh, Concave convex adaptive rejection sampling. J. Comput. Graph. Stat. 20(3), 670–691 (2011)Google Scholar
  12. 12.
    T.L. Griffiths, Z. Ghahramani, The indian buffet process: an introduction and review. J. Mach. Learn. Res. 12, 1185–1224 (2011)Google Scholar
  13. 13.
    H. Hirose, A. Todoroki, Random number generation for the generalized normal distribution using the modified adaptive rejection method. Int. Inf. Inst. 8(6), 829–836 (2005)Google Scholar
  14. 14.
    L. Holden, Adaptive chains. Technical Report Norwegian Computing Center (1998)Google Scholar
  15. 15.
    L. Holden, R. Hauge, M. Holden, Adaptive independent Metropolis-Hastings. Ann. Appl. Probab. 19(1), 395–413 (2009)Google Scholar
  16. 16.
    W. Hörmann, A rejection technique for sampling from T-concave distributions. ACM Trans. Math. Softw. 21(2), 182–193 (1995)Google Scholar
  17. 17.
    W. Hörmann, J. Leydold, G. Derflinger, Automatic Nonuniform Random Variate Generation (Springer, New York, 2003)Google Scholar
  18. 18.
    W. Hörmann, J. Leydold, G. Derflinger, Inverse transformed density rejection for unbounded monotone densities. Research Report Series, Department of Statistics and Mathematics (Economy and Business), Vienna University (2007)Google Scholar
  19. 19.
    Y. Huang, J. Zhang, P.M. Djurić, Bayesian detection for BLAST. IEEE Trans. Signal Process. 53(3), 1086–1096 (2005)Google Scholar
  20. 20.
    M.C. Jones, Distributions generated by transformation of scale using an extended cauchy-schlömilch transformation. Indian J. Stat. 72-A(2), 359–375 (2010)Google Scholar
  21. 21.
    J. Leydold, J. Janka, W. Hörmann, Variants of transformed density rejection and correlation induction, in Monte Carlo and Quasi-Monte Carlo Methods 2000 (Springer, Heidelberg, 2002), pp. 345–356Google Scholar
  22. 22.
    F. Liang, C. Liu, R. Caroll, Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples. Wiley Series in Computational Statistics (Wiley, Chichester, 2010)Google Scholar
  23. 23.
    L. Martino, Novel Schemes for Adaptive Rejection Sampling (Universidad Carlos III, Madrid, 2011)Google Scholar
  24. 24.
    L. Martino, Parsimonious adaptive rejection sampling. IET Electron. Lett. 53(16), 1115–1117 (2017)Google Scholar
  25. 25.
    L. Martino, F. Louzada, Adaptive rejection sampling with fixed number of nodes. Commun. Stat. Simul. Comput., 1–11 (2017, to appear)Google Scholar
  26. 26.
    L. Martino, J. Míguez, A generalization of the adaptive rejection sampling algorithm. Stat. Comput. 21, 633–647 (2010). doi: https://doi.org/10.1007/s11222-010-9197-9
  27. 27.
    L. Martino, J. Míguez, Generalized rejection sampling schemes and applications in signal processing. Signal Process. 90(11), 2981–2995 (2010)Google Scholar
  28. 28.
    L. Martino, J. Read, D. Luengo, Independent doubly adaptive rejection Metropolis sampling within Gibbs sampling, IEEE Trans. Signal Process. 63(12), 3123–3138 (2015)Google Scholar
  29. 29.
    L. Martino, R. Casarin, F. Leisen, D. Luengo, Adaptive independent sticky MCMC algorithms. EURASIP J. Adv. Signal Process. (2018, to appear)Google Scholar
  30. 30.
    R. Meyer, B. Cai, F. Perron, Adaptive rejection Metropolis sampling using Lagrange interpolation polynomials of degree 2. Comput. Stat. Data Anal. 52(7), 3408–3423 (2008)Google Scholar
  31. 31.
    J. Michel, The use of free energy simulations as scoring functions. PhD Thesis, University of Southampton (2006)Google Scholar
  32. 32.
    R. Neal, MCMC using Hamiltonian dynamics, Chap. 5, in Handbook of Markov Chain Monte Carlo ed. by S. Brooks, A. Gelman, G. Jones, X.-L. Meng (Chapman and Hall/CRC Press, Boca Raton, 2011)Google Scholar
  33. 33.
    J.G. Proakis, Digital Communications, 4th edn. (McGraw-Hill, Singapore, 2000)Google Scholar
  34. 34.
    Y.W. Teh, D. Görür, Z. Ghahramani, Stick-breaking construction for the indian buffet process, in Proceedings of the International Conference on Artificial Intelligence and Statistics (2007)Google Scholar
  35. 35.
    L. Tierney, Exploring posterior distributions using Markov Chains, in Computer Science and Statistics: Proceedings of IEEE 23rd Symposium on the Interface (1991), pp. 563–570Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Luca Martino
    • 1
  • David Luengo
    • 2
  • Joaquín Míguez
    • 1
  1. 1.Department of Signal Theory and CommunicationsCarlos III University of MadridMadridSpain
  2. 2.Department of Signal Theory and CommunicationsTechnical University of MadridMadridSpain

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