Bayesian Modular Inference for Copula Models with Potentially Misspecified Marginals

TL;DR

This paper introduces a Bayesian semi-modular inference method for copula models that allows flexible, individual control over marginal contributions, improving robustness against misspecification.

Contribution

It develops a novel copula semi-modular inference approach with Bayesian optimization for influence parameters, generalizing existing methods to handle varying marginal misspecification levels.

Findings

The method efficiently relaxes the combinatorial cut/uncut configurations.

Theoretical properties of the semi-modular posterior are established.

Applied to real data on equity volatility and bond yields, showing robustness.

Abstract

Copula models of multivariate data are popular because they allow separate specification of marginal distributions and the copula function. These components can be treated as inter-related modules in a modified Bayesian inference approach called ''cutting feedback'' that is robust to their misspecification. Recent work uses a two module approach, where all $d$ marginals form a single module, to robustify inference for the marginals against copula function misspecification, or vice versa. However, marginals can exhibit differing levels of misspecification, making it attractive to assign each its own module with an individual influence parameter controlling its contribution to a joint semi-modular inference (SMI) posterior. This generalizes existing two module SMI methods, which interpolate between cut and conventional posteriors using a single influence parameter. We develop a novel copula SMI method and select the influence parameters using Bayesian optimization. It provides an efficient continuous relaxation of the discrete optimization problem over $2^d$ cut/uncut configurations. We establish theoretical properties of the resulting semi-modular posterior and demonstrate the approach on simulated and real data. The real data application uses a skew-normal copula model of asymmetric dependence between equity volatility and bond yields, where robustifying copula estimation against marginal misspecification is strongly motivated.