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Statisticians often face the choice between using probability models or aparadigm defined by minimising a loss function. Both approaches are useful and,if the loss can be re-cast into a proper probability model, there are manytools to decide which model or loss is more appropriate for the observed data,in the sense of explaining the data's nature. However, when the loss leads toan improper model, there are no principled ways to guide this choice. Weaddress this task by combining the Hyv\"arinen score, which naturally targetsinfinitesimal relative probabilities, and general Bayesian updating, whichprovides a unifying framework for inference on losses and models. Specificallywe propose the H-score, a general Bayesian selection criterion and prove thatit consistently selects the (possibly improper) model closest to thedata-generating truth in Fisher's divergence. We also prove that an associatedH-posterior consistently learns optimal hyper-parameters featuring in lossfunctions, including a challenging tempering parameter in generalised Bayesianinference. As salient examples, we consider robust regression andnon-parametric density estimation where popular loss functions define impropermodels for the data and hence cannot be dealt with using standard modelselection tools. These examples illustrate advantages in robustness-efficiencytrade-offs and provide a Bayesian implementation for kernel density estimation,opening a new avenue for Bayesian non-parametrics.

General Bayesian Loss Function Selection and the use of Improper Models