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, and it is 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 2d 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.
The code has been tested with Python 3.12.2.
a) copula_smi.py: Contains code to train a Gaussian variational approximation to the SMI objective defined in the paper. The function takes an influence parameter gamma, the observed data, and functions to evaluate the statistical base model (copula and marginals) as input. Further documentation can be found in the file.
b) minimal_working_example.py: Contains a minimal working example on a single simulated dataset. It is recommended that you familiarize yourself with this example first.
c) Simulation study: simulation_study.py contains the code necessary to reproduce the simulation study and generate the figures presented in the manuscript.
d) Financial application: financial_application.py contains code to train the fully cut, conventional posterior, and optimal SMI models on the financial data. Plots are generated with financial_application_plots.py, which can be run after financial_application.py has run successfully. The preprocessed data are provided in yields.csv.
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