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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\u00e4rinen 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 \u210b-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 \u210b-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 enable Bayesian inference for kernel density estimation, opening a new avenue for Bayesian non-parametrics.<\/jats:p>","DOI":"10.1111\/rssb.12553","type":"journal-article","created":{"date-parts":[[2022,10,25]],"date-time":"2022-10-25T21:04:37Z","timestamp":1666731877000},"page":"1640-1665","source":"Crossref","is-referenced-by-count":18,"title":["General Bayesian Loss Function Selection and the use of Improper Models"],"prefix":"10.1093","volume":"84","author":[{"given":"Jack","family":"Jewson","sequence":"first","affiliation":[{"name":"Department of Business and Economics, Universitat Pompeu Fabra , Barcelona , Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Rossell","sequence":"additional","affiliation":[{"name":"Data Science Center, Barcelona School of Economics , Barcelona , Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,10,25]]},"reference":[{"issue":"3","key":"2023030906295695100_","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1093\/biomet\/85.3.549","article-title":"Robust and efficient estimation by minimising a density power divergence","volume":"85","author":"Basu","year":"1998","journal-title":"Biometrika"},{"issue":"2","key":"2023030906295695100_","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1080\/00401706.1974.10489171","article-title":"The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data","volume":"16","author":"Beaton","year":"1974","journal-title":"Technometrics"},{"issue":"4","key":"2023030906295695100_","doi-asserted-by":"crossref","first-page":"2025","DOI":"10.1093\/genetics\/162.4.2025","article-title":"Approximate Bayesian computation in population genetics","volume":"162","author":"Beaumont","year":"2002","journal-title":"Genetics"},{"key":"2023030906295695100_","volume-title":"Robust optimization for deep regression. 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