{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:31:17Z","timestamp":1760146277499,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,12]],"date-time":"2024-10-12T00:00:00Z","timestamp":1728691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Montana State University (Bozeman) Library","award":["DMR #1906383"],"award-info":[{"award-number":["DMR #1906383"]}]},{"name":"National Science Foundation","award":["DMR #1906383"],"award-info":[{"award-number":["DMR #1906383"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Empirical Bayes-based Methods (EBM) is an increasingly popular form of Objective Bayesianism (OB). It is identified in particular with the statistician Bradley Efron. The main aims of this paper are, first, to describe and illustrate its main features and, second, to locate its role by comparing it with two other statistical paradigms, Subjective Bayesianism (SB) and Evidentialism. EBM\u2019s main formal features are illustrated in some detail by schematic examples. The comparison between what Efron calls their underlying \u201cphilosophies\u201d is by way of a distinction made between confirmation and evidence. Although this distinction is sometimes made in the statistical literature, it is relatively rare and never to the same point as here. That is, the distinction is invariably spelled out intra- and not inter-paradigmatically solely in terms of one or the other accounts. The distinction made in this paper between confirmation and evidence is illustrated by two well-known statistical paradoxes: the base-rate fallacy and Popper\u2019s paradox of ideal evidence. The general conclusion reached is that each of the paradigms has a basic role to play and all are required by an adequate account of statistical inference from a technically informed and fine-grained philosophical perspective.<\/jats:p>","DOI":"10.3390\/e26100859","type":"journal-article","created":{"date-parts":[[2024,10,16]],"date-time":"2024-10-16T11:28:51Z","timestamp":1729078131000},"page":"859","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Empirical Bayes Methods, Evidentialism, and the Inferential Roles They Play"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6910-0291","authenticated-orcid":false,"given":"Samidha","family":"Shetty","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Montana State University, Bozeman, MT 59717, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-3233-8775","authenticated-orcid":false,"suffix":"Jr.","given":"Gordon","family":"Brittan","sequence":"additional","affiliation":[{"name":"Department of History and Philosophy, Montana State University, Bozeman, MT 59717, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Prasanta S.","family":"Bandyopadhyay","sequence":"additional","affiliation":[{"name":"Department of History and Philosophy, Montana State University, Bozeman, MT 59717, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1080\/01621459.2020.1762613","article-title":"Prediction, Estimation, and Attribution","volume":"115","author":"Efron","year":"2020","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_2","unstructured":"Bandyopadhyay, P. (2002, January 5\u20136). The Old Evidence Problem and Beyond. Proceedings of the Eastern Division Meeting, Boston, MA, USA."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bandyopadhyay, P., Brittan, G., and Taper, M. (2016). Belief, Evidence, and Uncertainty: Problems of Epistemic Inference, Springer Nature.","DOI":"10.1007\/978-3-319-27772-1"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Efron, B., and Hastie, T. (2016). Computer Age Statistical Inference: Algorithms, Evidence, and Data Science, Institute of Mathematical Statistics Monographs.","DOI":"10.1017\/CBO9781316576533"},{"key":"ref_5","unstructured":"Braithwaite, R.B. (1931). The Foundations of Mathematics and Other Logical Essays, Humanities Press."},{"key":"ref_6","unstructured":"De Finetti, B. (1937). Foresight: Its logical laws, its subjective sources. Breakthroughs in Statistics: Foundations and Basic Theory, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"437","DOI":"10.2307\/2564707","article-title":"Measuring Confirmation","volume":"99","author":"Christensen","year":"1999","journal-title":"J. Philos."},{"key":"ref_8","unstructured":"Psillos, S., and Curd, M. (2008). Confirmation. The Routledge Companion to Philosophy of Science, Routledge & Kegan Paul."},{"key":"ref_9","unstructured":"Howson, H., and Urbach, P. (2006). Scientific Reasoning: The Bayesian Approach, Open Court."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1093\/analys\/ant012","article-title":"Exhuming the No-Miracles Argument","volume":"73","author":"Howson","year":"2013","journal-title":"Analysis"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Joyce, J. (1999). The Foundations of Causal Decision Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511498497"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Springer, J., and Hartmann, S. (2019). Bayesian Philosophy of Science, Oxford University Press.","DOI":"10.1093\/oso\/9780199672110.001.0001"},{"key":"ref_13","unstructured":"Skyrms, B. (1986). Choice and Chance, Wadsworth Publishing Company."},{"key":"ref_14","unstructured":"Royall, R. (1997). Statistical Evidence: A Likelihood Paradigm, Chapman Hall."},{"key":"ref_15","unstructured":"Pagano, M., and Gauvreau, K. (2000). Principles of BioStatistics, Chapman Hall."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Taper, M., and Lele, S. (2004). Evidential Statistics and the Law of Likelihood. The Nature of Scientific Evidence, University of Chicago Press.","DOI":"10.7208\/chicago\/9780226789583.001.0001"},{"key":"ref_17","first-page":"362","article-title":"Measuring Confirmation and Evidence","volume":"17","author":"Ells","year":"2000","journal-title":"J. Philos."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Taper, M., and Lele, S. (2004). Why Likelihood?. The Nature of Scientific Evidence, University of Chicago Press.","DOI":"10.7208\/chicago\/9780226789583.001.0001"},{"key":"ref_19","unstructured":"Vickers, J. (1988). Chance and Structure, Clarendon Press."},{"key":"ref_20","unstructured":"Rosenkrantz, R. (1981). Inference, Method, and Decision, Reidel Publishing Company."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Jaynes, E. (2003). Probability Theory, Cambridge University Press. [3rd ed.].","DOI":"10.1017\/CBO9780511790423"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Williamson, J. (2005). Bayesian Nets and Causality, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198530794.001.0001"},{"key":"ref_23","first-page":"44","article-title":"Two Dogmas of Strong Objective Baysianism","volume":"24","author":"Bandyopadhay","year":"2010","journal-title":"Int. Stud. Philos. Sci."},{"key":"ref_24","first-page":"1","article-title":"Direct Inference and Probabilistic Accounts of Induction","volume":"54","author":"Williamson","year":"2022","journal-title":"J. Gen. Philos. Sci."},{"key":"ref_25","unstructured":"Paulos, J. (2001). Innumeracy: Mathematical Literacy and Its Consequences, Hill and Wang."},{"key":"ref_26","unstructured":"Fischer, R.A. (1959). Statistical Methods and Scientific Inference, Hafner Publishing Co.. [2nd ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1090\/S0273-0979-2012-01374-5","article-title":"A 250-year Argument: Belief, Behavior, and the Bootstrap","volume":"50","author":"Efron","year":"2010","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_28","first-page":"177","article-title":"Bayes, Oracle Bayes, and Empirical Bayes","volume":"34","author":"Efron","year":"2017","journal-title":"Stat. Sci."},{"key":"ref_29","unstructured":"Efron, B. (2021). Concepts and Methods, Stanford University."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Stein, C. (1956, January 26\u201331). Inadmissibility of the usual estimator for the mean of a multivariate distribution. Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Cambridge, UK.","DOI":"10.1525\/9780520313880-018"},{"key":"ref_31","unstructured":"James, W., and Stein, C. (July, January 20). Estimation with quadratic loss. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, California, CA, USA."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1038\/scientificamerican0577-119","article-title":"Stein\u2019s Paradox in Statistics","volume":"236","author":"Efron","year":"1977","journal-title":"Sci. Am."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1126\/science.1236536","article-title":"Bayes Theorem in the 21st century","volume":"340","author":"Efron","year":"2013","journal-title":"Science"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Lewis, D. (1980). A subjectivist\u2019s guide to objective chance. IFS: Conditionals, Belief, Decision, Chance and Time, Springer.","DOI":"10.1007\/978-94-009-9117-0_14"},{"key":"ref_35","unstructured":"Kahneman, D. (2011). Thinking Fast and Slow, Farrar, Strauss, and Giroux."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Popper, K. (1959). The Logic of Scientific Discovery, Hutchinson.","DOI":"10.1063\/1.3060577"},{"key":"ref_37","first-page":"145","article-title":"The future of indirect evidence","volume":"25","author":"Efron","year":"2010","journal-title":"Stat. Sci. A Rev. J. Inst. Math. Stat."},{"key":"ref_38","first-page":"92","article-title":"The Method of Multiple Working Hypotheses","volume":"15","author":"Chamberlain","year":"1900","journal-title":"Science"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1126\/science.146.3642.347","article-title":"Strong Inference","volume":"146","author":"Platt","year":"1964","journal-title":"Science"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/10\/859\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:12:05Z","timestamp":1760112725000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/10\/859"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,12]]},"references-count":39,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["e26100859"],"URL":"https:\/\/doi.org\/10.3390\/e26100859","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,10,12]]}}}