{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T05:57:21Z","timestamp":1791439041020,"version":"4.3.3"},"reference-count":39,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2024,11,21]],"date-time":"2024-11-21T00:00:00Z","timestamp":1732147200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"funder":[{"DOI":"10.13039\/100010663","name":"European Research Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"name":"European Union\u2019s Horizon 2020"},{"DOI":"10.13039\/100000001","name":"U.S. National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2203012"],"award-info":[{"award-number":["DMS-2203012"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,1,16]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>In recent years, there has been growing interest in statistical methods that exhibit robust performance under distribution changes between training and test data. While most of the related research focuses on point predictions with the squared error loss, this article turns the focus towards probabilistic predictions, which aim to comprehensively quantify the uncertainty of an outcome variable given covariates. Within a causality-inspired framework, we investigate the invariance and robustness of probabilistic predictions with respect to proper scoring rules. We show that arbitrary distribution shifts do not, in general, admit invariant and robust probabilistic predictions, in contrast to the setting of point prediction. We illustrate how to choose evaluation metrics and restrict the class of distribution shifts to allow for identifiability and invariance in the prototypical Gaussian heteroscedastic linear model. Motivated by these findings, we propose a method for obtaining invariant probabilistic predictions and study the consistency of the underlying parameters. Finally, we demonstrate the empirical performance of our proposed procedure via simulations and analysis of single-cell data.<\/jats:p>","DOI":"10.1093\/biomet\/asae063","type":"journal-article","created":{"date-parts":[[2024,11,21]],"date-time":"2024-11-21T16:29:38Z","timestamp":1732206578000},"source":"Crossref","is-referenced-by-count":1,"title":["Invariant probabilistic prediction"],"prefix":"10.1093","volume":"112","author":[{"given":"Alexander","family":"Henzi","sequence":"first","affiliation":[{"name":"Seminar for Statistics, ETH Z\u00fcrich , R\u00e4mistrasse 101 , 8092 Z\u00fcrich,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7801-4803","authenticated-orcid":false,"given":"Xinwei","family":"Shen","sequence":"additional","affiliation":[{"name":"Seminar for Statistics, ETH Z\u00fcrich , R\u00e4mistrasse 101 , 8092 Z\u00fcrich,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Law","sequence":"additional","affiliation":[{"name":"Seminar for Statistics, ETH Z\u00fcrich , R\u00e4mistrasse 101 , 8092 Z\u00fcrich,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1782-6015","authenticated-orcid":false,"given":"Peter","family":"B\u00fchlmann","sequence":"additional","affiliation":[{"name":"Seminar for Statistics, ETH Z\u00fcrich , R\u00e4mistrasse 101 , 8092 Z\u00fcrich,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,11,21]]},"reference":[{"key":"2025030418471118800_asae063-B1","author":"Ai","year":"2024"},{"key":"2025030418471118800_asae063-B2","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1080\/01621459.1996.10476902","article-title":"Identification of causal effects using instrumental variables","volume":"91","author":"Angrist","year":"1996","journal-title":"J. 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