Regression Copulas for Multivariate Responses

TL;DR

This paper introduces a new Bayesian distributional regression model using copula processes for multivariate responses, enabling accurate marginal distribution estimation and improved predictive performance in large datasets.

Contribution

It develops a novel regression copula model with a multivariate horseshoe prior and efficient variational inference, enhancing multivariate response modeling and calibration.

Findings

Outperforms benchmark methods in electricity price forecasting

Achieves well-calibrated marginal distributions in applications

Enables scalable Bayesian inference for large datasets

Abstract

We propose a novel distributional regression model for a multivariate response vector based on a copula process over the covariate space. It uses the implicit copula of a Gaussian multivariate regression, which we call a ``regression copula''. To allow for large covariate vectors their coefficients are regularized using a novel multivariate extension of the horseshoe prior. Bayesian inference and distributional predictions are evaluated using efficient variational inference methods, allowing application to large datasets. An advantage of the approach is that the marginal distributions of the response vector can be estimated separately and accurately, resulting in predictive distributions that are marginally-calibrated. Two substantive applications of the methodology highlight its efficacy in multivariate modeling. The first is the econometric modeling and prediction of half-hourly regional Australian electricity prices. Here, our approach produces more accurate distributional forecasts than leading benchmark methods. The second is the evaluation of multivariate posteriors in likelihood-free inference (LFI) of a model for tree species abundance data, extending a previous univariate regression copula LFI method. In both applications, we demonstrate that our new approach exhibits a desirable marginal calibration property.