Computationally Efficient Bayesian Estimation of High Dimensional 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 $2^{J}$ terms, with $J$ the number of discrete variables. Currently, data augmentation methods are used to carry out inference for discrete copula and, in practice, the computation becomes infeasible when $J$ is large. Our article proposes two new fast Bayesian approaches for estimating high dimensional 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 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 Monte Carlo and randomised quasi Monte Carlo methods can be used with both approaches to reduce the variability of the estimate of the likelihood, and hence enable us to carry out Bayesian inference for high values of $J$ for some classes of copulas where the computation was previously too expensive. Our article also introduces {\em a correlated quasi random number pseudo marginal} approach into the literature. The methodology is illustrated through several real and simulated data examples.