{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T17:45:11Z","timestamp":1770745511650,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,15]],"date-time":"2020-06-15T00:00:00Z","timestamp":1592179200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["17K14233"],"award-info":[{"award-number":["17K14233"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["18K12757"],"award-info":[{"award-number":["18K12757"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Although linear regression models are fundamental tools in statistical science, the estimation results can be sensitive to outliers. While several robust methods have been proposed in frequentist frameworks, statistical inference is not necessarily straightforward. We here propose a Bayesian approach to robust inference on linear regression models using synthetic posterior distributions based on \u03b3-divergence, which enables us to naturally assess the uncertainty of the estimation through the posterior distribution. We also consider the use of shrinkage priors for the regression coefficients to carry out robust Bayesian variable selection and estimation simultaneously. We develop an efficient posterior computation algorithm by adopting the Bayesian bootstrap within Gibbs sampling. The performance of the proposed method is illustrated through simulation studies and applications to famous datasets.<\/jats:p>","DOI":"10.3390\/e22060661","type":"journal-article","created":{"date-parts":[[2020,6,15]],"date-time":"2020-06-15T12:16:57Z","timestamp":1592223417000},"page":"661","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Robust Bayesian Regression with Synthetic Posterior Distributions"],"prefix":"10.3390","volume":"22","author":[{"given":"Shintaro","family":"Hashimoto","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hiroshima University, Hiroshima 739-8521, Japan"}]},{"given":"Shonosuke","family":"Sugasawa","sequence":"additional","affiliation":[{"name":"Center for Spatial Information Science, The University of Tokyo, Chiba 277-8568, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,15]]},"reference":[{"key":"ref_1","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"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2053","DOI":"10.1016\/j.jmva.2008.02.004","article-title":"Robust parameter estimation with a small bias against heavy contamination","volume":"99","author":"Fujisawa","year":"2008","journal-title":"J. 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