Bayesian Inference for Regression Copulas
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We propose a new semi-parametric distributional regression smoother for continuous data, which 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 non-parametric functions of the covariates modeled using Bayesian splines. By integrating out the spline coefficients, we derive an implicit copula that captures dependence as a smooth non-parametric function of the covariates, which we call a regression copula. We derive some of its properties, and show that the entire distribution - including the mean and variance - of the response from the copula model are also smooth non-parametric functions of the covariates. Even though the implicit copula cannot be expressed in closed form, we estimate it efficiently using both Hamiltonian Monte Carlo and variational Bayes methods. Using four real data examples, we illustrate the efficacy of these estimators, and show the properties and advantages of the copula model, for implicit copulas of dimension up to 40,981. The approach produces predictive densities of the response that are locally adaptive with respect to the covariates, and are more accurate than those from benchmark methods in every case.
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