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Monte Carlo and Approximation Methods

High-Dimensional Copula Variational Approximation Through Transformation

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Pages 729-743 | Received 20 Feb 2019, Accepted 04 Mar 2020, Published online: 20 Apr 2020

References

  • Aas, K., Czado, C., Frigessi, A., and Bakken, H. (2009), “Pair-Copula Constructions of Multiple Dependence,” Insurance: Mathematics and Economics, 44, 182–198. DOI: 10.1016/j.insmatheco.2007.02.001.
  • Archer, E., Park, I. M., Buesing, L., Cunningham, J., and Paninski, L. (2016), “Black Box Variational Inference for State Space Models,” arXiv no. 1511.07367.
  • Azzalini, A., and Capitanio, A. (2003), “Distributions Generated by Perturbation of Symmetry With Emphasis on a Multivariate Skew t-Distribution,” Journal of the Royal Statistical Society, Series B, 65, 367–389. DOI: 10.1111/1467-9868.00391.
  • Azzalini, A., and Dalla Valle, A. (1996), “The Multivariate Skew-Normal Distribution,” Biometrika, 83, 715–726. DOI: 10.1093/biomet/83.4.715.
  • Blei, D. M., Kucukelbir, A., and McAuliffe, J. D. (2017), “Variational Inference: A Review for Statisticians,” Journal of the American Statistical Association, 112, 859–877. DOI: 10.1080/01621459.2017.1285773.
  • Bottou, L. (2010), “Large-Scale Machine Learning With Stochastic Gradient Descent,” in Proceedings of the 19th International Conference on Computational Statistics (COMPSTAT’2010), eds. Y. Lechevallier and G. Saporta, Springer, pp.177–187.
  • Carpenter, B., Gelman, A., Hoffman, M., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A. (2017), “Stan: A Probabilistic Programming Language,” Journal of Statistical Software, 76, 1–32. DOI: 10.18637/jss.v076.i01.
  • Challis, E., and Barber, D. (2013), “Gaussian Kullback–Leibler Approximate Inference,” The Journal of Machine Learning Research, 14, 2239–2286.
  • Demarta, S., and McNeil, A. J. (2005), “The t Copula and Related Copulas,” International Statistical Review, 73, 111–129. DOI: 10.1111/j.1751-5823.2005.tb00254.x.
  • Dinh, L., Sohl-Dickstein, J., and Bengio, S. (2016), “Density Estimation Using Real NVP,” arXiv no. 1605.08803.
  • Elidan, G. (2010), “Copula Bayesian Networks,” in Advances in Neural Information Processing Systems (Vol. 23), eds. J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel, and A. Culotta, La Jolla, CA: NIPS Foundation, pp. 559–567.
  • Fang, H.-B., Fang, K.-T., and Kotz, S. (2002), “The Meta-Elliptical Distributions With Given Marginals,” Journal of Multivariate Analysis, 82, 1–16. DOI: 10.1006/jmva.2001.2017.
  • Genest, C., and Nešlehová, J. (2007), “A Primer on Copulas for Count Data,” ASTIN Bulletin: The Journal of the IAA, 37, 475–515. DOI: 10.2143/AST.37.2.2024077.
  • Genton, M. G. (2004), Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality, Boca Raton, FL: CRC Press.
  • Guo, F., Wang, X., Broderick, T., and Dunson, D. B. (2016), “Boosting Variational Inference,” arXiv no. 1611.05559.
  • Han, S., Liao, X., Dunson, D., and Carin, L. (2016), “Variational Gaussian Copula Inference,” in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research (Vol. 51), eds. A. Gretton and C. C. Robert, Cadiz, Spain: PMLR, pp. 829–838.
  • Headrick, T. C., Kowalchuk, R. K., and Sheng, Y. (2008), “Parametric Probability Densities and Distribution Functions for Tukey g-and-h Transformations and Their Use for Fitting Data,” Applied Mathematical Sciences, 2, 449–462.
  • Hosmer, D. W., Lemeshow, S., and Sturdivant, R. X. (2013), Applied Logistic Regression (3rd ed.), Hoboken, NJ: Wiley.
  • Huszár, F. (2017), “Variational Inference Using Implicit Distributions,” arXiv no. 1702.08235.
  • Kingma, D. P., and Welling, M. (2014), “Auto-Encoding Variational Bayes,” arXiv no. 1312.6114.
  • Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., and Blei, D. M. (2017), “Automatic Differentiation Variational Inference,” Journal of Machine Learning Research, 18, 430–474.
  • Liu, Q., and Wang, D. (2016), “Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm,” in Advances in Neural Information Processing Systems, pp. 2378–2386.
  • Loaiza-Maya, R., and Smith, M. S. (2019), “Variational Bayes Estimation of Discrete-Margined Copula Models With Application to Time Series,” Journal of Computational and Graphical Statistics, 28, 523–539. DOI: 10.1080/10618600.2018.1562936.
  • McNeil, A. J., Frey, R., and Embrechts, P. (2005), Quantitative Risk Management: Concepts, Techniques and Tools, Princeton Series in Finance.
  • Miller, A. C., Foti, N., and Adams, R. P. (2016), “Variational Boosting: Iteratively Refining Posterior Approximations,” arXiv no. 1611.06585.
  • Murray, J. S., Dunson, D. B., Carin, L., and Lucas, J. E. (2013), “Bayesian Gaussian Copula Factor Models for Mixed Data,” Journal of the American Statistical Association, 108, 656–665. DOI: 10.1080/01621459.2012.762328.
  • Nelsen, R. B. (2006), An Introduction to Copulas, Springer Series in Statistics, Secaucus, NJ:Springer-Verlag New York, Inc.
  • Oh, D. H., and Patton, A. J. (2017), “Modeling Dependence in High Dimensions With Factor Copulas,” Journal of Business & Economic Statistics, 35, 139–154. DOI: 10.1080/07350015.2015.1062384.
  • Ong, V. M.-H., Nott, D. J., and Smith, M. S. (2018), “Gaussian Variational Approximation With a Factor Covariance Structure,” Journal of Computational and Graphical Statistics, 27, 465–478. DOI: 10.1080/10618600.2017.1390472.
  • Opper, M., and Archambeau, C. (2009), “The Variational Gaussian Approximation Revisited,” Neural Computation, 21, 786–792. DOI: 10.1162/neco.2008.08-07-592.
  • Ormerod, J. T. (2011), “Skew-Normal Variational Approximations for Bayesian Inference,” Technical Report, School of Mathematics and Statistics, University of Sydney.
  • Ormerod, J. T., and Wand, M. P. (2010), “Explaining Variational Approximations,” The American Statistician, 64, 140–153. DOI: 10.1198/tast.2010.09058.
  • Peters, G., Chen, W., and Gerlach, R. (2016), “Estimating Quantile Families of Loss Distributions for Non-Life Insurance Modelling via L-Moments,” Risks, 4, 14. DOI: 10.3390/risks4020014.
  • Quiroz, M., Nott, D. J., and Kohn, R. (2018), “Gaussian Variational Approximation for High-Dimensional State Space Models,” arXiv no. 1801.07873.
  • Ranganath, R., Tran, D., and Blei, D. (2016), “Hierarchical Variational Models,” in Proceedings of the 33rd International Conference on Machine Learning, Proceedings of Machine Learning Research (Vol. 48), M. F. Balcan and K. Q. Weinberger, New York, NY: PMLR, pp. 324–333.
  • Rezende, D. J., and Mohamed, S. (2015), “Variational Inference With Normalizing Flows,” in Proceedings of the 32nd International Conference on Machine Learning, Proceedings of Machine Learning Research (Vol. 37), eds. F. Bach and D. Blei, Lille, France: PMLR, pp. 1530–1538.
  • Rezende, D. J., Mohamed, S., and Wierstra, D. (2014), “Stochastic Backpropagation and Approximate Inference in Deep Generative Models,” in Proceedings of the 31st International Conference on Machine Learning, Proceedings of Machine Learning Research (Vol. 32), eds. E. P. Xing and T. Jebara, Bejing, China: PMLR, pp. 1278–1286.
  • Robbins, H., and Monro, S. (1951), “A Stochastic Approximation Method,” The Annals of Mathematical Statistics, 22, 400–407. DOI: 10.1214/aoms/1177729586.
  • Rue, H., Martino, S., and Chopin, N. (2009), “Approximate Bayesian Inference for Latent Gaussian Models by Using Integrated Nested Laplace Approximations,” Journal of the Royal Statistical Society, Series B, 71, 319–392. DOI: 10.1111/j.1467-9868.2008.00700.x.
  • Salimans, T., and Knowles, D. A. (2013), “Fixed-Form Variational Posterior Approximation Through Stochastic Linear Regression,” Bayesian Analysis, 8, 837–882. DOI: 10.1214/13-BA858.
  • Seeger, M. (2000), “Bayesian Model Selection for Support Vector Machines, Gaussian Processes and Other Kernel Classifiers,” in Advances in Neural Information Processing Systems (Vol. 12), eds. S. A. Solla, T. K. Leen, and K. Müller, Cambridge, MA: MIT Press, pp. 603–609.
  • Smith, M. S. (2015), “Copula Modelling of Dependence in Multivariate Time Series,” International Journal of Forecasting, 31, 815–833. DOI: 10.1016/j.ijforecast.2014.04.003.
  • Smith, M. S., Gan, Q., and Kohn, R. J. (2012), “Modelling Dependence Using Skew t Copulas: Bayesian Inference and Applications,” Journal of Applied Econometrics, 27, 500–522. DOI: 10.1002/jae.1215.
  • Smith, M. S., and Khaled, M. (2012), “Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation,” Journal of the American Statistical Association, 107, 290–303. DOI: 10.1080/01621459.2011.644501.
  • Smith, M. S., and Maneesoonthorn, W. (2018), “Inversion Copulas From Nonlinear State Space Models With an Application to Inflation Forecasting,” International Journal of Forecasting, 34, 389–407. DOI: 10.1016/j.ijforecast.2018.01.002.
  • Song, X.-K. P. (2000), “Multivariate Dispersion Models Generated From Gaussian Copula,” Scandinavian Journal of Statistics, 27, 305–320.
  • Spantini, A., Bigoni, D., and Marzouk, Y. (2018), “Inference via Low-Dimensional Couplings,” Journal of Machine Learning Research, 19, 2639–2709.
  • Su, Q., Liao, X., Chen, C., and Carin, L. (2016), “Nonlinear Statistical Learning With Truncated Gaussian Graphical Models,” in ICML, JMLR Workshop and Conference Proceedings (Vol. 48), JMLR.org, pp. 1948–1957.
  • Tan, L. S. L., and Nott, D. J. (2018), “Gaussian Variational Approximation With Sparse Precision Matrices,” Statistics and Computing, 28, 259–275. DOI: 10.1007/s11222-017-9729-7.
  • Titsias, M., and Lázaro-Gredilla, M. (2014), “Doubly Stochastic Variational Bayes for Non-Conjugate Inference,” in, Proceedings of the 31st International Conference on Machine Learning, Proceedings of Machine Learning Research (Vol. 32), eds. E. P. Xing and T. Jebara, Beijing, China: PMLR, pp. 1971–1979.
  • Tran, D., Blei, D. M., and Airoldi, E. M. (2015), “Copula Variational Inference,” in Advances in Neural Information Processing Systems 28: Annual Conference on Neural Information Processing Systems 2015, December 7–12, 2015, Montreal, Quebec, Canada, pp. 3564–3572.
  • Tukey, T. W. (1977), “Modern Techniques in Data Analysis,” in NSP-Sponsored Regional Research Conference at Southeastern Massachusetts University, North Dartmouth, MA.
  • Xu, M., Quiroz, M., Kohn, R., and Sisson, S. A. (2018), “On Some Variance Reduction Properties of the Reparameterization Trick,” arXiv no. 1809.10330.
  • Yeo, I.-K., and Johnson, R. A. (2000), “A New Family of Power Transformations to Improve Normality or Symmetry,” Biometrika, 87, 954–959. DOI: 10.1093/biomet/87.4.954.
  • Yoshiba, T. (2018), “Maximum Likelihood Estimation of Skew-t Copulas With Its Applications to Stock Returns,” Journal of Statistical Computation and Simulation, 88, 2489–2506. DOI: 10.1080/00949655.2018.1469631.
  • Zeiler, M. D. (2012), “ADADELTA: An Adaptive Learning Rate Method,” arXiv no. 1212.5701.

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