Differentially Private Bayesian Inference for Gaussian Copula Correlations

Gaussian copulas are widely used to estimate multivariate distributions and relationships. We present algorithms for estimating Gaussian copula correlations that ensure differential privacy. We first convert data values into sets of two-way tables of counts above and below marginal medians. We then add noise to these counts to satisfy differential privacy. We utilize the one-to-one correspondence between the true counts and the copula correlation to estimate a posterior distribution of the copula correlation given the noisy counts, marginalizing over the distribution of the underlying true counts using a composite likelihood. We also present an alternative, maximum likelihood approach for point estimation. Using simulation studies, we compare these methods to extant methods in the literature for computing differentially private copula correlations.