Advertisement

Computationally efficient Bayesian estimation of high-dimensional Archimedean copulas with discrete and mixed margins

  • D. Gunawan
  • M.-N. TranEmail author
  • K. Suzuki
  • J. Dick
  • R. Kohn
Article
  • 33 Downloads

Abstract

Estimating copulas with discrete marginal distributions is challenging, especially in high dimensions, because computing the likelihood contribution of each observation requires evaluating \(2^{J}\) terms, with J the number of discrete variables. Our article focuses on the estimation of Archimedean copulas, for example, Clayton and Gumbel copulas. Currently, data augmentation methods are used to carry out inference for discrete copulas and, in practice, the computation becomes infeasible when J is large. Our article proposes two new fast Bayesian approaches for estimating high-dimensional Archimedean copulas with discrete margins, or a combination of discrete and continuous margins. Both methods are based on recent advances in Bayesian methodology that work with an unbiased estimate of the likelihood rather than the likelihood itself, and our key observation is that we can estimate the likelihood of a discrete Archimedean copula unbiasedly with much less computation than evaluating the likelihood exactly or with current simulation methods that are based on augmenting the model with latent variables. The first approach builds on the pseudo-marginal method that allows Markov chain Monte Carlo simulation from the posterior distribution using only an unbiased estimate of the likelihood. The second approach is based on a variational Bayes approximation to the posterior and also uses an unbiased estimate of the likelihood. We show that the two new approaches enable us to carry out Bayesian inference for high values of J for the Archimedean copulas where the computation was previously too expensive. The methodology is illustrated through several real and simulated data examples.

Keywords

Markov chain Monte Carlo Correlated pseudo-marginal Metropolis–Hastings Variational Bayes Archimedean copula 

Notes

Supplementary material

11222_2018_9846_MOESM1_ESM.pdf (98 kb)
Supplementary material 1 (pdf 97 KB)

References

  1. Andrieu, C., Roberts, G.: The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Stat. 37, 697–725 (2009)MathSciNetCrossRefGoogle Scholar
  2. Beaumont, M.A.: Estimation of population growth or decline in genetically monitored populations. Genetics 164, 1139–1160 (2003)Google Scholar
  3. Deligiannidis, G., Doucet, A., Pitt, M.: The correlated pseudo-marginal method. J. R. Stat. Soc. B 80, 839–870 (2018)CrossRefGoogle Scholar
  4. Doucet, A., Pitt, M., Deligiannidis, G., Kohn, R.: Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. Biometrika 102(2), 295–313 (2015)MathSciNetCrossRefGoogle Scholar
  5. Flury, T., Shephard, N.: Bayesian inference based only on simulated likelihood: particle filter analysis of dynamic economic models. Econom. Theory 27(5), 933–956 (2011)MathSciNetCrossRefGoogle Scholar
  6. Garthwaite, P.H., Fan, Y., Sisson, S.A.: Adaptive optimal scaling of Metropolis–Hastings algorithms using the Robbins–Monro process. Commun. Stat. Theory Methods 45(17), 5098–5111 (2016)MathSciNetCrossRefGoogle Scholar
  7. Hofert, M.: Sampling Archimedian copulas. Comput. Stat. Data Anal. 52(12), 5163–5174 (2008)MathSciNetCrossRefGoogle Scholar
  8. Hofert, M., Machler, M., Mcneil, A.J.: Likelihood inference for Archimedian copulas in high dimensions under known margins. J. Multivar. Anal. 110, 133–150 (2012)CrossRefGoogle Scholar
  9. Kingma, D.P., Welling, M.: Auto-encoding Variational Bayes. In: Proceedings of the 2nd International Conference on Learning Representations (ICLR). https://arxiv.org/abs/1312.6114 (2014)
  10. Murray, J.S., Dunson, D.B., Carin, L., Lucas, J.: Bayesian Gaussian copula factor models for mixed data. J. Am. Stat. Assoc. 108(502), 656–665 (2013)MathSciNetCrossRefGoogle Scholar
  11. Ormerod, J.T., Wand, M.P.: Explaining variational approximations. Am. Stat. 64(2), 140–153 (2010)MathSciNetCrossRefGoogle Scholar
  12. Pakman, A., Paninski, L.: Exact Hamiltonian Monte Carlo for truncated multivariate Gaussian. J. Comput. Graph. Stat. 23(2), 518–542 (2014)MathSciNetCrossRefGoogle Scholar
  13. Panagiotelis, A., Czado, C., Joe, H.: Pair copula constructions for multivariate discrete data. J. Am. Stat. Assoc. 107(499), 1063–1072 (2012)MathSciNetCrossRefGoogle Scholar
  14. Panagiotelis, A., Czado, C., Joe, H., Stober, J.: Model selection for discrete regular vine copulas. Comput. Stat. Data Anal. 106, 138–152 (2017)MathSciNetCrossRefGoogle Scholar
  15. Pitt, M., Chan, D., Kohn, R.: Efficient Bayesian inference for Gaussian copula regression models. Biometrika 93(3), 537–554 (2006)MathSciNetCrossRefGoogle Scholar
  16. Pitt, M.K., Silva, R.S., Giordani, P., Kohn, R.: On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econom. 171(2), 134–151 (2012)MathSciNetCrossRefGoogle Scholar
  17. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Stat. 22(3), 400–407 (1951)MathSciNetCrossRefGoogle Scholar
  18. Roberts, G.O., Gelman, A., Gilks, W.R.: Weak convergence and optimal scaling of random walk Metropolis–Hastings. Ann. Appl. Probab. 7, 110–120 (1997)MathSciNetCrossRefGoogle Scholar
  19. Sherlock, C., Thiery, A., Roberts, G., Rosenthal, J.: On the efficiency of pseudo marginal random walk Metropolis algorithm. Ann. Stat. 43(1), 238–275 (2015)MathSciNetCrossRefGoogle Scholar
  20. Sklar, A.: Fonctions de Répartition à n Dimensions et Leurs Marges [Distributional Functions ton Dimensions and Their Margins]. vol. 8, pp. 229–231. Publications de l’Institut Statistique de l’Université de Paris (1959)Google Scholar
  21. Smith, M., Khaled, M.A.: Estimation of copula models with discrete margins via Bayesian data augmentation. J. Am. Stat. Assoc. 107(497), 290–303 (2012)MathSciNetCrossRefGoogle Scholar
  22. Tran, M. N., Kohn, R., Quiroz, M., Villani, M.: Block-wise pseudo marginal Metropolis–Hastings. Preprint arXiv:1603.02485v2 (2016)
  23. Tran, M.-N., Nott, D., Kohn, R.: Variational Bayes with intractable likelihood. J. Comput. Graph. Stat. 26(4), 873–882 (2017)MathSciNetCrossRefGoogle Scholar
  24. Trivedi, P., Zimmer, D.: Copula modeling: an introduction for practitioners. Found. Trends Econom. 1(1), 1–111 (2005)CrossRefGoogle Scholar
  25. Ware, J.E., Snow, K.K., Kolinski, M., Gandeck, B.: SF-36 health survey manual and interpretation guide. The Health Institute New England Medical Centre, Boston (1993)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.UNSW Business SchoolUniversity of New South WalesSydneyAustralia
  2. 2.The University of Sydney Business SchoolSydneyAustralia
  3. 3.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

Personalised recommendations