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In many applications, however, prior distributions are introduced merely as instruments to facilitate computation, rather than as representations of genuine subjective belief. Consequently, relying on standard Bayesian justifications for inferential procedures becomes conceptually ungrounded. In this paper, we recommend evaluating finite\u2010sample performance over repeated sampling of data and parameters as an alternative justification for \u201cpragmatic Bayes.\u201d We demonstrate a key vulnerability in the usual posterior\u2010based inference: when analysts' chosen prior distribution mismatches the true parameter\u2010generating process, Bayesian inference can be misleading in the long run. Given that this true process is rarely known in practice, we propose a safer alternative: calibrating Bayesian credible regions to achieve frequentist validity. This latter criterion is stronger and guarantees validity of Bayesian inference regardless of the underlying parameter\u2010generating mechanism. To solve the calibration problem in practice, we propose a novel stochastic approximation algorithm. Monte Carlo experiments are conducted and reported, in which we observe that uncalibrated Bayesian inference can be liberal under certain parameter\u2010generating scenarios, whereas our calibrated solution consistently maintain validity. We also illustrate the proposed calibration procedure using a real\u2010data example involving location\u2010scale regression.<\/jats:p>","DOI":"10.1111\/bmsp.70065","type":"journal-article","created":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T09:15:02Z","timestamp":1785402902000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Calibrating Bayesian inference"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3559-688X","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"first","affiliation":[{"name":"University of Maryland  College Park Maryland USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5164-9430","authenticated-orcid":false,"given":"Jonathan P.","family":"Williams","sequence":"additional","affiliation":[{"name":"North Carolina State University  Raleigh North Carolina USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4164-0173","authenticated-orcid":false,"given":"Jan","family":"Hannig","sequence":"additional","affiliation":[{"name":"The University of North Carolina at Chapel Hill  Chapel Hill North Carolina USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,7,30]]},"reference":[{"key":"e_1_2_13_2_1","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830244"},{"key":"e_1_2_13_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/100802529"},{"key":"e_1_2_13_4_1","doi-asserted-by":"publisher","DOI":"10.2307\/2347792"},{"key":"e_1_2_13_5_1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2018.0565"},{"key":"e_1_2_13_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4286-2"},{"key":"e_1_2_13_7_1","doi-asserted-by":"publisher","DOI":"10.1142\/13640"},{"key":"e_1_2_13_8_1","doi-asserted-by":"publisher","DOI":"10.1214\/14\u2010BA915"},{"key":"e_1_2_13_9_1","doi-asserted-by":"publisher","DOI":"10.1214\/ss\/1177013238"},{"key":"e_1_2_13_10_1","volume-title":"Mathematical statistics: Basic ideas and selected topics, volume i","author":"Bickel P. 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