Approximate Bayesian Conditional Copulas

TL;DR

This paper introduces Bayesian nonparametric methods to estimate functionals of dependence in copula models, accommodating covariates and avoiding copula selection, with validation through simulations and real-world applications.

Contribution

It develops novel Bayesian nonparametric approaches for approximating dependence functionals, addressing the challenge of copula model selection and incorporating covariate effects.

Findings

Methods effectively estimate dependence functionals in simulations.

Approaches successfully applied to civil engineering and astrophysics data.

Bayesian methods provide uncertainty quantification for dependence measures.

Abstract

Copula models are flexible tools to represent complex structures of dependence for multivariate random variables. According to Sklar's theorem (Sklar, 1959), any d-dimensional absolutely continuous density can be uniquely represented as the product of the marginal distributions and a copula function which captures the dependence structure among the vector components. In real data applications, the interest of the analyses often lies on specific functionals of the dependence, which quantify aspects of it in a few numerical values. A broad literature exists on such functionals, however extensions to include covariates are still limited. This is mainly due to the lack of unbiased estimators of the copula function, especially when one does not have enough information to select the copula model. Recent advances in computational methodologies and algorithms have allowed inference in the presence of complicated likelihood functions, especially in the Bayesian approach, whose methods, despite being computationally intensive, allow us to better evaluate the uncertainty of the estimates. In this work, we present several Bayesian methods to approximate the posterior distribution of functionals of the dependence, using nonparametric models which avoid the selection of the copula function. These methods are compared in simulation studies and in two realistic applications, from civil engineering and astrophysics.