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

Better together? Statistical learning in models made of modules