Epistemically robust selection of fitted models

Abstract: Fitting models to data is an important part of the practice of science, made almost ubiquitous by advances in machine learning. Very often however, fitted solutions are not unique, but form an ensemble of candidate models -- qualitatively different, yet with comparable quantitative performance. One then needs a criterion which can select the best candidate models, or at least falsify (reject) the worst ones. Because standard statistical approaches to model selection rely on assumptions which are usually invalid in scientific contexts, they tend to be overconfident, rejecting models based on little more than statistical noise. The ideal objective for fitting models is generally considered to be the risk: this is the theoretical average loss of a model (assuming unlimited data). In this work we develop a nonparametric method for estimating, for each candidate model, the epistemic uncertainty on its risk: in other words we associate to each model a distribution of scores which accounts for expected modelling errors. We then propose that a model falsification criterion should mirror established experimental practice: a falsification result should be accepted only if it is reproducible across experimental variations. The strength of this approach is illustrated using examples from physics and neuroscience.