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Estimating copulas with discrete marginal distributions is challenging,especially in high dimensions, because computing the likelihood contribution ofeach observation requires evaluating $2^{J}$ terms, with $J$ the number ofdiscrete variables. Currently, data augmentation methods are used to carry outinference for discrete copula and, in practice, the computation becomesinfeasible when $J$ is large. Our article proposes two new fast Bayesianapproaches for estimating high dimensional copulas with discrete margins, or acombination of discrete and continuous margins. Both methods are based onrecent advances in Bayesian methodology that work with an unbiased estimate ofthe likelihood rather than the likelihood itself, and our key observation isthat we can estimate the likelihood of a discrete copula unbiasedly with muchless computation than evaluating the likelihood exactly or with currentsimulation methods that are based on augmenting the model with latentvariables. The first approach builds on the pseudo marginal method that allowsMarkov chain Monte Carlo simulation from the posterior distribution using onlyan unbiased estimate of the likelihood. The second approach is based on aVariational Bayes approximation to the posterior and also uses an unbiasedestimate of the likelihood. We show that Monte Carlo and randomised quasi MonteCarlo methods can be used with both approaches to reduce the variability of theestimate of the likelihood, and hence enable us to carry out Bayesian inferencefor high values of $J$ for some classes of copulas where the computation waspreviously too expensive. Our article also introduces {\em a correlated quasirandom number pseudo marginal} approach into the literature. The methodology isillustrated through several real and simulated data examples.

Computationally Efficient Bayesian Estimation of High Dimensional Copulas with Discrete and Mixed Margins