{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:29:09Z","timestamp":1760239749377,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,27]],"date-time":"2020-12-27T00:00:00Z","timestamp":1609027200000},"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":["JP19K14597","JP17K14233"],"award-info":[{"award-number":["JP19K14597","JP17K14233"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum \u03b3-divergence estimator is well-known to work well in estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In the objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribution based on the \u03b3-divergence. In particular, we show that the proposed priors are approximately robust under the condition on the contamination distribution without assuming any conditions on the contamination ratio. Some simulation studies are also presented.<\/jats:p>","DOI":"10.3390\/e23010029","type":"journal-article","created":{"date-parts":[[2020,12,27]],"date-time":"2020-12-27T20:04:58Z","timestamp":1609099498000},"page":"29","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Default Priors for Robust Bayesian Estimation with Divergences"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5057-5927","authenticated-orcid":false,"given":"Tomoyuki","family":"Nakagawa","sequence":"first","affiliation":[{"name":"Department of Information Sciences, Tokyo University of Science, Chiba 278-8510, Japan"}]},{"given":"Shintaro","family":"Hashimoto","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hiroshima University, Hiroshima 739-8521, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Huber, J., and Ronchetti, E.M. (2009). Robust Statistics, Wiley. [2nd ed.].","DOI":"10.1002\/9780470434697"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Basu, A., Shioya, H., and Park, C. (2011). Statistical Inference: The Minimum Distance Approach, Chapman & Hall.","DOI":"10.1201\/b10956"},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1093\/biomet\/88.3.865","article-title":"A comparison of related density-based minimum divergence estimators","volume":"88","author":"Jones","year":"2001","journal-title":"Biometrika"},{"key":"ref_5","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. Multivar. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.jmva.2017.07.012","article-title":"Robust sparse Gaussian graphical modeling","volume":"161","author":"Hirose","year":"2016","journal-title":"J. Multivar. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kawashima, T., and Fujisawa, H. (2017). Robust and sparse regression via \u03b3-divergence. Entropy, 19.","DOI":"10.3390\/e19110608"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Hirose, K., and Masuda, H. (2018). Robust relative error estimation. Entropy, 20.","DOI":"10.3390\/e20090632"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1111\/rssb.12158","article-title":"A general framework for updating belief distributions","volume":"78","author":"Bissiri","year":"2016","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1007\/s11749-014-0360-z","article-title":"Bayesian model robustness via disparities","volume":"23","author":"Hooker","year":"2014","journal-title":"Test"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s10463-014-0499-0","article-title":"Robust Bayes estimation using the density power divergence","volume":"68","author":"Ghosh","year":"2016","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1080\/03610926.2018.1543765","article-title":"Robust Bayesian inference via \u03b3-divergence","volume":"49","author":"Nakagawa","year":"2020","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Jewson, J., Smith, J.Q., and Holmes, C. (2018). Principles of Bayesian inference using general divergence criteria. Entropy, 20.","DOI":"10.3390\/e20060442"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hashimoto, S., and Sugasawa, S. (2020). Robust Bayesian regression with synthetic posterior distributions. Entropy, 22.","DOI":"10.3390\/e22060661"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1111\/j.2517-6161.1979.tb01066.x","article-title":"Reference posterior distributions for Bayesian inference","volume":"41","author":"Bernardo","year":"1979","journal-title":"J. R. Stat. Soc. Ser. B Methodol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/s13171-011-0012-2","article-title":"Moment matching priors","volume":"73","author":"Ghosh","year":"2011","journal-title":"Sankhya A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1007\/s11749-018-0597-z","article-title":"Objective Bayesian inference with proper scoring rules","volume":"28","author":"Mameli","year":"2019","journal-title":"Test"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2278","DOI":"10.3150\/13-BEJ557","article-title":"Affine invariant divergences associated with proper composite scoring rules and their applications","volume":"20","author":"Kanamori","year":"2014","journal-title":"Bernoulli"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s10463-009-0226-4","article-title":"A general divergence criterion for prior selection","volume":"63","author":"Ghosh","year":"2011","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1214\/14-BA862","article-title":"On divergence measures leading to Jeffreys and other reference priors","volume":"9","author":"Liu","year":"2014","journal-title":"Bayesian Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.jspi.2020.11.007","article-title":"Reference priors via \u03b1-divergence for a certain non-regular model in the presence of a nuisance parameter","volume":"213","author":"Hashimoto","year":"2021","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.jspi.2019.03.009","article-title":"Moment matching priors for non-regular models","volume":"203","author":"Hashimoto","year":"2019","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Robert, C.P., and Casella, G. (2004). Monte Carlo Statistical Methods, Springer.","DOI":"10.1007\/978-1-4757-4145-2"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Serfling, R. (1980). Approximation Theorems of Mathematical Statistics, Wiley.","DOI":"10.1002\/9780470316481"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1080\/00949650412331299120","article-title":"Choosing a robustness tuning parameter","volume":"75","author":"Warwick","year":"2005","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1093\/biomet\/asz075","article-title":"Robust empirical Bayes small area estimation with density power divergence","volume":"107","author":"Sugasawa","year":"2020","journal-title":"Biometrika"},{"key":"ref_27","unstructured":"Basak, S., Basu, A., and Jones, M. (2020). On the \u2018optimal\u2019 density power divergence tuning parameter. J. Appl. Stat., 1\u201321."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1093\/biomet\/asv014","article-title":"Robust estimation under heavy contamination using unnormalized models","volume":"102","author":"Kanamori","year":"2015","journal-title":"Biometrika"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:46:43Z","timestamp":1760179603000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/1\/29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,27]]},"references-count":28,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["e23010029"],"URL":"https:\/\/doi.org\/10.3390\/e23010029","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,12,27]]}}}