Abstract
Bayesian evidence combination often involves models with multiple parametric modules. When one module is misspecified, its influence can be reduced by cutting information flow—this leads to Cut Models. Semi-modular inference (SMI) generalizes this by introducing a learning rate, loss function, and Gibbs posterior to allow partial information flow between modules. Estimating the learning rate and loss hyperparameters is challenging, as they can't be learned jointly with other parameters due to model misspecification. We consider settings with a pseudo-optimal learning rate and use held-out data to infer its posterior. Under certain conditions, this posterior concentrates on the pseudo-optimal value, enabling principled hyperparameter estimation. Experiments show that SMI posteriors outperform both Cut-model and standard Bayesian approaches on test data.