Title: Improper models for data analysis
Authors: David Rossell - Universitat Pompeu Fabra (Spain) [presenting]
Jack Jewson - Universitat Pompeu Fabra and Barcelona Graduate School of Economics (Spain)
Abstract: Statisticians often face the choice between using probability models or a paradigm 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 many tools to decide which model or loss is more appropriate to explain the data's nature. However, when the loss leads to an improper model, there are no principled ways to guide this choice. We address this task by combining the Hyvrinen score, which naturally targets infinitesimal relative probabilities, and general Bayesian updating, which provides a unifying framework for inference on losses and models. Specifically, we propose the H-score, a general Bayesian selection criterion and prove that it consistently selects the (possibly improper) model closest to the data-generating truth in Fisher's divergence. We also prove that an associated H-posterior consistently learns optimal hyper-parameters featuring in loss functions, including a challenging tempering parameter in generalised Bayes / Gibbs posteriors / PAC Bayes. As examples, we consider robust regression and non-parametric density estimation, where popular loss functions define improper models for the data. We hence cannot be dealt with using standard model selection tools. These examples illustrate advantages in robustness-efficiency trade-offs and provide a Bayesian implementation for kernel density estimation, opening a new avenue for Bayesian non-parametrics.