Abstract:  Specifying a Bayesian prior is notoriously difficult for complex models such as neural networks. Reasoning about parameters is made challenging by the high-dimensionality and over-parameterization of the space. Priors that seem benign and uninformative can have unintuitive and detrimental effects on a model’s predictions. For this reason, we propose predictive complexity priors: a prior that is defined by comparing the model’s predictions to those of a reference model. Although originally defined on the model outputs, we transfer the prior to the model parameters via a change of variables. The traditional Bayesian workflow can then proceed as usual. We apply our predictive complexity prior to modern machine learning tasks such as reasoning over neural network depth and sharing of statistical strength for few-shot learning.