  This work considers Bayesian inference under misspecification for complex
statistical models comprised of simpler submodels, referred to as modules, that
are coupled together. Such ``multi-modular" models often arise when combining
information from different data sources, where there is a module for each data
source. When some of the modules are misspecified, the challenges of Bayesian
inference under misspecification can sometimes be addressed by using ``cutting
feedback" methods, which modify conventional Bayesian inference by limiting the
influence of unreliable modules. Here we investigate cutting feedback methods
in the context of generalized posterior distributions, which are built from
arbitrary loss functions, and present novel findings on their behaviour. We
make three main contributions. First, we describe how cutting feedback methods
can be defined in the generalized Bayes setting, and discuss the appropriate
scaling of the loss functions for different modules to each other and the
prior. Second, we derive a novel result about the large sample behaviour of the
posterior for a given module's parameters conditional on the parameters of
other modules. This formally justifies the use of conditional Laplace
approximations, which provide better approximations of conditional posterior
distributions compared to conditional distributions from a Laplace
approximation of the joint posterior. Our final contribution leverages the
large sample approximations of our second contribution to provide convenient
diagnostics for understanding the sensitivity of inference to the coupling of
the modules, and to implement a new semi-modular posterior approach for
conducting robust Bayesian modular inference. The usefulness of the methodology
is illustrated in several benchmark examples from the literature on cut model
inference.
