General Bayesian Loss Function Selection and the use of Improper Models

TL;DR

This paper introduces the H-score, a Bayesian criterion for selecting models based on loss functions, especially when models are improper, and demonstrates its effectiveness in robust regression and density estimation.

Contribution

It proposes the H-score, a Bayesian model selection method that handles improper models and optimally learns hyper-parameters, advancing model choice in non-standard settings.

Findings

H-score consistently selects models closest to the true data-generating process.

The associated H-posterior reliably learns optimal hyper-parameters.

Application to density estimation demonstrates Bayesian non-parametric capabilities.

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 for the observed data, in the sense of explaining 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 Hyv\"arinen 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 Bayesian inference. As salient examples, we consider robust regression and non-parametric density estimation where popular loss functions define improper models for the data and 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.