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Bayesian statistical inference loses predictive optimality when generativemodels are misspecified.  Working within an existing coherent loss-based generalisation of Bayesianinference, we show existing Modular/Cut-model inference is coherent, and writedown a new family of Semi-Modular Inference (SMI) schemes, indexed by aninfluence parameter, with Bayesian inference and Cut-models as special cases.We give a meta-learning criterion and estimation procedure to choose theinference scheme. This returns Bayesian inference when there is nomisspecification.  The framework applies naturally to Multi-modular models. Cut-model inferenceallows directed information flow from well-specified modules to misspecifiedmodules, but not vice versa. An existing alternative power posterior methodgives tunable but undirected control of information flow, improving predictionin some settings. In contrast, SMI allows tunable and directed information flowbetween modules.  We illustrate our methods on two standard test cases from the literature anda motivating archaeological data set.

Semi-Modular Inference: enhanced learning in multi-modular models by tempering the influence of components