  In modern applications, statisticians are faced with integrating
heterogeneous data modalities relevant for an inference, prediction, or
decision problem. In such circumstances, it is convenient to use a graphical
model to represent the statistical dependencies, via a set of connected
"modules", each relating to a specific data modality, and drawing on specific
domain expertise in their development. In principle, given data, the
conventional statistical update then allows for coherent uncertainty
quantification and information propagation through and across the modules.
However, misspecification of any module can contaminate the estimate and update
of others, often in unpredictable ways. In various settings, particularly when
certain modules are trusted more than others, practitioners have preferred to
avoid learning with the full model in favor of approaches that restrict the
information propagation between modules, for example by restricting propagation
to only particular directions along the edges of the graph. In this article, we
investigate why these modular approaches might be preferable to the full model
in misspecified settings. We propose principled criteria to choose between
modular and full-model approaches. The question arises in many applied
settings, including large stochastic dynamical systems, meta-analysis,
epidemiological models, air pollution models,
pharmacokinetics-pharmacodynamics, and causal inference with propensity scores.
