# A New Rejection Sampling Method for Truncated Multivariate Gaussian Random Variables Restricted to Convex Sets

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 163)

## Abstract

Statistical researchers have shown increasing interest in generating truncated multivariate normal distributions. In this paper, we only assume that the acceptance region is convex and we focus on rejection sampling. We propose a new algorithm that outperforms crude rejection method for the simulation of truncated multivariate Gaussian random variables. The proposed algorithm is based on a generalization of Von Neumann’s rejection technique which requires the determination of the mode of the truncated multivariate density function. We provide a theoretical upper bound for the ratio of the target probability density function over the proposal probability density function. The simulation results show that the method is especially efficient when the probability of the multivariate normal distribution of being inside the acceptance region is low.

## Keywords

Truncated Gaussian vector Rejection sampling Monte Carlo method

## Notes

### Acknowledgments

This work has been conducted within the frame of the ReDice Consortium, gathering industrial (CEA, EDF, IFPEN, IRSN, Renault) and academic (Ecole des Mines de Saint-Etienne, INRIA, and the University of Bern) partners around advanced methods for Computer Experiments. The authors wish to thank Olivier Roustant (EMSE), Laurence Grammont (ICJ, Lyon 1) and Yann Richet (IRSN, Paris) for helpful discussions, as well as the anonymous reviewers for constructive comments and the participants of MCQMC2014 conference.

## References

1. 1.
Botts, C.: An accept-reject algorithm for the positive multivariate normal distribution. Comput. Stat. 28(4), 1749–1773 (2013)
2. 2.
Breslaw, J.: Random sampling from a truncated multivariate normal distribution. Appl. Math. Lett. 7(1), 1–6 (1994)
3. 3.
Casella, G., George, E.I.: Explaining the Gibbs sampler. Am. Stat. 46(3), 167–174 (1992)
4. 4.
Chopin, N.: Fast simulation of truncated Gaussian distributions. Stat. Comput. 21(2), 275–288 (2011)
5. 5.
Da Veiga, S., Marrel, A.: Gaussian process modeling with inequality constraints. Annales de la faculté des sciences de Toulouse 21(3), 529–555 (2012)
6. 6.
Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986)Google Scholar
7. 7.
Ellis, N., Maitra, R.: Multivariate Gaussian simulation outside arbitrary ellipsoids. J. Comput. Graph. Stat. 16(3), 692–708 (2007)
8. 8.
Emery, X., Arroyo, D., Peláez, M.: Simulating large Gaussian random vectors subject to inequality constraints by Gibbs sampling. Math. Geosci. 1–19 (2013)Google Scholar
9. 9.
Freulon, X., Fouquet, C.: Conditioning a Gaussian model with inequalities. In: Soares, A. (ed.) Geostatistics Tróia ’92, Quantitative Geology and Geostatistics, vol. 5, pp. 201–212. Springer, Netherlands (1993)Google Scholar
10. 10.
Gelfand, A.E., Smith, A.F.M., Lee, T.M.: Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling. J. Am. Stat. Assoc. 87(418), 523–532 (1992)
11. 11.
Geweke, J.: Exact inference in the inequality constrained normal linear regression model. J. Appl. Econom. 1(2), 127–141 (1986)
12. 12.
Geweke, J.: Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints and the evaluation of constraint probabilities. In: Proceedings of the 23rd Symposium on the Interface Computing Science and Statistics, pp. 571–578 (1991)Google Scholar
13. 13.
Gilks, W.R., Wild, P.: Adaptive rejection sampling for Gibbs sampling. J. R. Stat. Soc. Series C (Applied Statistics) 41(2), 337–348 (1992)
14. 14.
Goldfarb, D., Idnani, A.: A numerically stable dual method for solving strictly convex quadratic programs. Math. Progr. 27(1), 1–33 (1983)
15. 15.
Griffiths, W.E.: A Gibbs sampler for the parameters of a truncated multivariate normal distribution. Department of Economics - Working Papers Series 856, The University of Melbourne (2002)Google Scholar
16. 16.
Hörmann, W., Leydold, J., Derflinger, G.: Automatic Nonuniform Random Variate Generation. Statistics and Computing. Springer, Berlin (2004)
17. 17.
Kotecha, J.H., Djuric, P.: Gibbs sampling approach for generation of truncated multivariate Gaussian random variables. IEEE Int. Conf. Acoust. Speech Signal Process. 3, 1757–1760 (1999)Google Scholar
18. 18.
Laud, P.W., Damien, P., Shively, T.S.: Sampling some truncated distributions via rejection algorithms. Commun. Stat. - Simulation Comput. 39(6), 1111–1121 (2010)
19. 19.
Li, Y., Ghosh, S.K.: Efficient sampling method for truncated multivariate normal and student t-distribution subject to linear inequality constraints. http://www.stat.ncsu.edu/information/library/papers/mimeo2649_Li.pdf
20. 20.
Maatouk, H., Bay, X.: Gaussian process emulators for computer experiments with inequality constraints (2014). https://hal.archives-ouvertes.fr/hal-01096751
21. 21.
Martino, L., Miguez, J.: An adaptive accept/reject sampling algorithm for posterior probability distributions. In: IEEE/SP 15th Workshop on Statistical Signal Processing, SSP ’09, pp. 45–48 (2009)Google Scholar
22. 22.
Martino, L., Miguez, J.: A novel rejection sampling scheme for posterior probability distributions. In: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing ICASSP, pp. 2921–2924 (2009)Google Scholar
23. 23.
Philippe, A., Robert, C.P.: Perfect simulation of positive Gaussian distributions. Stat. Comput. 13(2), 179–186 (2003)
24. 24.
Robert, C.P.: Simulation of truncated normal variables. Stat. Comput. 5(2) (1995)Google Scholar
25. 25.
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods. Springer, Berlin (2004)Google Scholar
26. 26.
Rodriguez-Yam, G., Davis, R.A., Scharf, L.L.: Efficient Gibbs sampling of truncated multivariate normal with application to constrained linear regression (2004). http://www.stat.columbia.edu/~rdavis/papers/CLR.pdf
27. 27.
Von Neumann, J.: Various techniques used in connection with random digits. J. Res. Nat. Bur. Stand. 12, 36–38 (1951)Google Scholar
28. 28.
Jun-wu YU, G.l.T.: Efficient algorithms for generating truncated multivariate normal distributions. Acta Mathematicae Applicatae Sinica, English Series 27(4), 601 (2011)Google Scholar