  In copula models the marginal distributions and copula function are specified
separately. We treat these as two modules in a modular Bayesian inference
framework, and propose conducting modified Bayesian inference by "cutting
feedback". Cutting feedback limits the influence of potentially misspecified
modules in posterior inference. We consider two types of cuts. The first limits
the influence of a misspecified copula on inference for the marginals, which is
a Bayesian analogue of the popular Inference for Margins (IFM) estimator. The
second limits the influence of misspecified marginals on inference for the
copula parameters by using a pseudo likelihood of the ranks to define the cut
model. We establish that if only one of the modules is misspecified, then the
appropriate cut posterior gives accurate uncertainty quantification
asymptotically for the parameters in the other module. Computation of the cut
posteriors is difficult, and new variational inference methods to do so are
proposed. The efficacy of the new methodology is demonstrated using both
simulated data and a substantive multivariate time series application from
macroeconomic forecasting. In the latter, cutting feedback from misspecified
marginals to a 1096 dimension copula improves posterior inference and
predictive accuracy greatly, compared to conventional Bayesian inference.
