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

Estimating copulas with discrete marginal distributions is challenging, especially in high dimensions, because computing the likelihood contribution of each observation requires evaluating $$2^{J}$$2J 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.