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Copula Based Factorization in Bayesian Multivariate Infinite Mixture Models

  • Martin Burda
  • Artem Prokhorov

Bayesian nonparametric models based on infinite mixtures of density kernels have been recently gaining in popularity due to their flexibility and feasibility of implementation even in complicated modeling scenarios. In economics, they have been particularly useful in estimating nonparametric distributions of latent variables. However, these models have been rarely applied in more than one dimension. Indeed, the multivariate case suffers from the curse of dimensionality, with a rapidly increasing number of parameters needed to jointly characterize each mixing component. In this paper, we propose a factorization scheme for nonparametric mixture models whereby each marginal dimension in the mixing parameter space is modeled separately, linked by a nonparametric random copula function. Specifically, we consider nonparametric univariate Gaussian mixtures for the marginals and a multivariate random Bernstein polynomial copula for the link function, under Dirichlet process priors. We show that this scheme leads to an improvement in the precision of a density estimate in finite samples, providing a suitable tool for applications in higher dimensions. We derive weak posterior consistency of the copula-based mixing scheme for general kernel types under high-level conditions, and strong posterior consistency for the specific Bernstein-Gaussian mixture model.

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Paper provided by University of Toronto, Department of Economics in its series Working Papers with number tecipa-473.

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Date of creation: 28 Jan 2013
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Handle: RePEc:tor:tecipa:tecipa-473
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  1. Zheng, Yanbing, 2011. "Shape restriction of the multi-dimensional Bernstein prior for density functions," Statistics & Probability Letters, Elsevier, vol. 81(6), pages 647-651, June.
  2. Ausin, M. Concepcion & Lopes, Hedibert F., 2010. "Time-varying joint distribution through copulas," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2383-2399, November.
  3. Conley, Timothy G. & Hansen, Christian B. & McCulloch, Robert E. & Rossi, Peter E., 2008. "A semi-parametric Bayesian approach to the instrumental variable problem," Journal of Econometrics, Elsevier, vol. 144(1), pages 276-305, May.
  4. Axel Tenbusch, 1994. "Two-dimensional Bernstein polynomial density estimators," Metrika, Springer, vol. 41(1), pages 233-253, December.
  5. Burda, Martin & Harding, Matthew & Hausman, Jerry, 2008. "A Bayesian mixed logit-probit model for multinomial choice," Journal of Econometrics, Elsevier, vol. 147(2), pages 232-246, December.
  6. Chib, Siddhartha & Hamilton, Barton H., 2002. "Semiparametric Bayes analysis of longitudinal data treatment models," Journal of Econometrics, Elsevier, vol. 110(1), pages 67-89, September.
  7. Kim Jin Gyo & Menzefricke Ulrich & Feinberg Fred M., 2004. "Assessing Heterogeneity in Discrete Choice Models Using a Dirichlet Process Prior," Review of Marketing Science, De Gruyter, vol. 2(1), pages 1-41, January.
  8. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(03), pages 535-562, June.
  9. Jensen, Mark J. & Maheu, John M., 2010. "Bayesian semiparametric stochastic volatility modeling," Journal of Econometrics, Elsevier, vol. 157(2), pages 306-316, August.
  10. Rodríguez, Abel & Dunson, David B. & Gelfand, Alan E., 2010. "Latent Stick-Breaking Processes," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 647-659.
  11. Paolo Giordani & Xiuyan Mun & Robert Kohn, 2012. "Efficient Estimation of Covariance Matrices using Posterior Mode Multiple Shrinkage," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 11(1), pages 154-192, December.
  12. Keisuke Hirano, 2002. "Semiparametric Bayesian Inference in Autoregressive Panel Data Models," Econometrica, Econometric Society, vol. 70(2), pages 781-799, March.
  13. Yanbing Zheng & Jun Zhu & Anindya Roy, 2010. "Nonparametric Bayesian inference for the spectral density function of a random field," Biometrika, Biometrika Trust, vol. 97(1), pages 238-245.
  14. Michael Pitt & David Chan & Robert Kohn, 2006. "Efficient Bayesian inference for Gaussian copula regression models," Biometrika, Biometrika Trust, vol. 93(3), pages 537-554, September.
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