## Bayesian Inference for Regression Copulas

We propose a new semiparametric distributional regression smoother that is based on a copula decomposition of the joint distribution of the vector of response values. The copula is high-dimensional and constructed by inversion of a pseudo regression, where the conditional mean and variance are semiparametric functions of covariates modeled using regularized basis functions. By integrating out the basis coefficients, an implicit copula process on the covariate space is obtained, which we call a “regression copula.” We combine this with a nonparametric margin to define a copula model, where the entire distribution—including the mean and variance—of the response is a smooth semiparametric function of the covariates. The copula is estimated using both Hamiltonian Monte Carlo and variational Bayes; the latter of which is scalable to high dimensions. Using real data examples and a simulation study, we illustrate the efficacy of these estimators and the copula model. In a substantive example, we estimate the distribution of half-hourly electricity spot prices as a function of demand and two time covariates using radial bases and horseshoe regularization. The copula model produces distributional estimates that are locally adaptive with respect to the covariates, and predictions that are more accurate than those from benchmark models. Supplementary materials for this article are available online.