Estimation of Copula Models With Discrete Margins via Bayesian Data Augmentation

Abstract

Estimation of copula models with discrete margins can be difficult beyond the bivariate case. We show how this can be achieved by augmenting the likelihood with latent variables, and computing inference using the resulting augmented posterior. To evaluate this we propose two efficient Markov chain Monte Carlo sampling schemes. One generates the latent variables as a block using a Metropolis-Hasting step with a proposal that is close to its target distribution, the other generates them one at a time. Our method applies to all parametric copulas where the conditional copula functions can be evaluated, not just elliptical copulas as in much previous work. Moreover, the copula parameters can be estimated joint with any marginal parameters, and Bayesian selection ideas employed. We establish the effectiveness of the estimation method by modeling consumer behavior in online retail using Archimedean and Gaussian copulas. The example shows that elliptical copulas can be poor at modeling dependence in discrete data, just as they can be in the continuous case. To demonstrate the potential in higher dimensions we estimate 16 dimensional D-vine copulas for a longitudinal model of usage of a bicycle path in the city of Melbourne, Australia. The estimates reveal an interesting serial dependence structure that can be represented in a parsimonious fashion using Bayesian selection of independence pair-copula components. Finally, we extend our results and method to the case where some margins are discrete and others continuous. Supplemental materials for the article are also available online.

Mixed Density Based Copula Likelihood

“…In this paper, we propesed a flexible yet computationally simple copula based alternative. Unlike earlier proposals such as those of Pitt et al (2006), Smith andKhaled (2012) or Hoff (2007), our method is entirely classical and uses straightforward maximum likelihood methods for esstimation rather than a Bayesian sampling approach. The latter typically is computationally more expensive.…”

“…During the estimation, these coordinates are considered latent and are integrated out. Smith and Khaled (2012) extend the previous methodology to the case of an arbitrary copula function. Both papers use Markov Chain Monte Carlo (MCMC) techniques.…”

Bayesian Nonparametric Inference for a Multivariate Copula Function

“…This real data set is about the number of bicycles traveling down the Main Yarra South Bank bike path in Melbourne, analyzed by Smith and Khaled (2012 Triple peak times, 07:01-08:00, 09:01-10:00 and 16:01-17:00, are used here to illustrate our mixture model, can be extended to higher dimensional cases in a straightforward manner. The real data are transformed according to the strategy outlined in Section 1.…”

Vine copulas for mixed data : multi-view clustering for mixed data beyond meta-Gaussian dependencies

“…Parameter Estimation for Discrete Vines For vines with discrete margins, Smith and Khaled (2012) propose an MCMC inference algorithm which uses a data augmentation approach to compute the probability mass function (PMF). It is extensible to mixed data, but requires, for a M-dimensional vine, O(2 M ) computations per sampling step.…”

“…The Gaussian Copula (with meta-Gaussian dependencies) on the left cannot model data with asymmetric dependencies shown on the right there are no existing techniques to fit vine copulas on mixed data and the challenge lies mainly in parameter inference. Two previous approaches, both for only discrete (not mixed) features, require expensive estimation of marginals: one by Panagiotelis et al (2012) and another by Smith and Khaled (2012). The latter can be extended to mixed data but their MCMC algorithm requires computations that are exponential in data dimensions per sampling step, making it practically infeasible.…”