Bayesian Modular Inference for Copula Models with Potentially Misspecified Marginals

2026-03-11

AI Summary

This study addresses the challenge of simultaneous misspecification of marginal distributions and the copula function in copula modeling by proposing a multi-module semi-modular inference (SMI) approach. The method introduces independent modules for each marginal distribution, each governed by its own influence parameter, and employs continuous relaxation to circumvent discrete search over exponentially many cut configurations. Integrated with Bayesian optimization for automatic hyperparameter tuning, the framework adaptively accommodates varying degrees of misspecification across margins. Theoretical analysis and empirical evaluations demonstrate that the proposed method achieves notable robustness and superior performance on both synthetic data and real-world financial datasets—such as modeling the asymmetric dependence between equity volatility and bond yields using a skew-normal copula.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

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.

Problem

• copula models
• marginal misspecification
• Bayesian inference
• semi-modular inference
• robustness

Innovation

• Semi-Modular Inference
• Copula Models
• Bayesian Optimization
• Modular Bayesian Inference
• Marginal Misspecification