{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T07:30:04Z","timestamp":1791531004932,"version":"4.3.4"},"reference-count":33,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2007,11,30]],"date-time":"2007-11-30T00:00:00Z","timestamp":1196380800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2008,4]]},"abstract":"<jats:p>We consider the properties of the highest posterior probability model in a linear regression setting. Under a spike and slab hierarchy we find that although highest posterior model selection is total risk consistent, it possesses hidden undesirable properties. One such property is a marked underfitting in finite samples, a phenomenon well noted for Bayesian information criterion (BIC) related procedures but not often associated with highest posterior model selection. Another concern is the substantial effect the prior has on model selection. We employ a rescaling of the hierarchy and show that the resulting rescaled spike and slab models mitigate the effects of underfitting because of a perfect cancellation of a BIC-like penalty term. Furthermore, by drawing upon an equivalence between the highest posterior model and the median model, we find that the effect of the prior is less influential on model selection, as long as the underlying true model is sparse. Nonsparse settings are, however, problematic. Using the posterior mean for variable selection instead of posterior inclusion probabilities avoids these issues.<\/jats:p>","DOI":"10.1017\/s026646660808016x","type":"journal-article","created":{"date-parts":[[2007,11,30]],"date-time":"2007-11-30T12:07:45Z","timestamp":1196424465000},"page":"377-403","source":"Crossref","is-referenced-by-count":8,"title":["AN IN-DEPTH LOOK AT HIGHEST POSTERIOR MODEL SELECTION"],"prefix":"10.1017","volume":"24","author":[{"given":"Tanujit","family":"Dey","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hemant","family":"Ishwaran","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. Sunil","family":"Rao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2007,11,30]]},"reference":[{"key":"S026646660808016X_ref016","doi-asserted-by":"crossref","unstructured":"Ishwaran, H. & J. Rao (2005a)Spike and slab gene selection for multigroup microarray data.Journal of the American Statistical Association 100,764\u2013780.","DOI":"10.1198\/016214505000000051"},{"key":"S026646660808016X_ref030","doi-asserted-by":"crossref","unstructured":"Stone, M. (1977b)Asymptotics for and against cross-validation.Biometrika 64,29\u201335.","DOI":"10.1093\/biomet\/64.1.29"},{"key":"S026646660808016X_ref008","doi-asserted-by":"crossref","unstructured":"Clyde, M. , G. Parmigiani , & B. Vidakovic (1998)Multiple shrinkageand subset selection in wavelets.Biometrika 85,391\u2013402.","DOI":"10.1093\/biomet\/85.2.391"},{"key":"S026646660808016X_ref028","doi-asserted-by":"crossref","unstructured":"Smith, M. & R. Kohn (1996)Nonparametric regression using Bayesian variable selection.Journal of Econometrics 75,317\u2013344.","DOI":"10.1016\/0304-4076(95)01763-1"},{"key":"S026646660808016X_ref004","doi-asserted-by":"crossref","unstructured":"Bernardo, J. & A. Smith (1994)Bayesian Theory.Wiley.","DOI":"10.1002\/9780470316870"},{"key":"S026646660808016X_ref027","doi-asserted-by":"crossref","unstructured":"Shibata, R. (1976)Selection of the order of an autoregressive model by Akaike's information.Biometrika 63,117\u2013126.","DOI":"10.1093\/biomet\/63.1.117"},{"key":"S026646660808016X_ref031","doi-asserted-by":"crossref","unstructured":"Yang, Y. (2005)Can the strengths of AIC and BIC be shared? A conflict between model identification and regression estimation.Biometrika 92,937\u2013950.","DOI":"10.1093\/biomet\/92.4.937"},{"key":"S026646660808016X_ref010","doi-asserted-by":"crossref","unstructured":"George, E. & R. McCulloch (1993)Variable selection via Gibbs sampling.Journal of the American Statistical Association 88,881\u2013889.","DOI":"10.1080\/01621459.1993.10476353"},{"key":"S026646660808016X_ref013","doi-asserted-by":"crossref","unstructured":"Hannan, E. & B. Quinn (1979)The determination of the order of an autoregression.Journal of the Royal Statistical Society, Series B 41,190\u2013195.","DOI":"10.1111\/j.2517-6161.1979.tb01072.x"},{"key":"S026646660808016X_ref029","doi-asserted-by":"crossref","unstructured":"Stone, M. (1977a)An asymptotic equivalence of choice of model by cross-validation and Akaike's criterion.Journal of the Royal Statistical Society, Series B 39,44\u201347.","DOI":"10.1111\/j.2517-6161.1977.tb01603.x"},{"key":"S026646660808016X_ref002","doi-asserted-by":"crossref","unstructured":"Atkinson, A. (1978)Posterior probabilities for choosing a regression model.Biometrika 65,39\u201348.","DOI":"10.1093\/biomet\/65.1.39"},{"key":"S026646660808016X_ref021","doi-asserted-by":"crossref","unstructured":"Nishii, R. (1984)Asymptotic properties of criteria for selection of variables in multiple regression.Annals of Statistics 12,758\u2013765.","DOI":"10.1214\/aos\/1176346522"},{"key":"S026646660808016X_ref025","doi-asserted-by":"crossref","unstructured":"Shao, J. (1993)Linear model selection by cross-validation.Journal of the American Statistical Association 88,486\u2013494.","DOI":"10.1080\/01621459.1993.10476299"},{"key":"S026646660808016X_ref018","unstructured":"Leeb, H. & B. 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McCulloch (1997)Approaches for Bayesian variable selection.Statistica Sinica 7,339\u2013373."},{"key":"S026646660808016X_ref012","doi-asserted-by":"crossref","unstructured":"Geweke, J. & R. Meese (1981)Estimating regression models of finite but unknown order.International Economic Review 22,55\u201370.","DOI":"10.2307\/2526135"},{"key":"S026646660808016X_ref024","unstructured":"Serfling, R. (2002)Approximation Theorems of Mathematical Statistics.Wiley."},{"key":"S026646660808016X_ref003","doi-asserted-by":"crossref","unstructured":"Barbieri, M. & J. Berger (2004)Optimal predictive model selection.Annals of Statistics 32,870\u2013897.","DOI":"10.1214\/009053604000000238"},{"key":"S026646660808016X_ref019","doi-asserted-by":"crossref","unstructured":"Leeb, H. & B. P\u00f6tscher (2005)Model selection and inference: Facts and fiction.Econometric Theory 21,21\u201359.","DOI":"10.1017\/S0266466605050036"},{"key":"S026646660808016X_ref017","doi-asserted-by":"crossref","unstructured":"Ishwaran, H. & J. Rao (2005b)Spike and slab variable selection: Frequentist and Bayesian strategies.Annals of Statistics 33,730\u2013773.","DOI":"10.1214\/009053604000001147"},{"key":"S026646660808016X_ref023","doi-asserted-by":"crossref","unstructured":"Schwarz, G. (1978)Estimating the dimension of a model.Annals of Statistics 6,461\u2013464.","DOI":"10.1214\/aos\/1176344136"},{"key":"S026646660808016X_ref022","doi-asserted-by":"crossref","unstructured":"Rao, C. & Y. Wu (1989)A strongly consistent procedure for model selection in a regression problem.Biometrika 76,369\u2013374.","DOI":"10.1093\/biomet\/76.2.369"},{"key":"S026646660808016X_ref005","doi-asserted-by":"crossref","unstructured":"Breiman, L. (1992)The little bootstrap and other methods for dimensionality selection in regression: X-fixed prediction error.Journal of the American Statistical Association 87,738\u2013754.","DOI":"10.1080\/01621459.1992.10475276"},{"key":"S026646660808016X_ref033","doi-asserted-by":"crossref","unstructured":"Zhang, P. (1993b)On the convergence rate of model selection criteria.Communications in Statistical Theory and Methods 22,2765\u20132775.","DOI":"10.1080\/03610929308831184"},{"key":"S026646660808016X_ref014","doi-asserted-by":"crossref","unstructured":"Hoeting, J.A. , D. Madigan , A.E. Raftery , & C.T. Volinsky (1999)Bayesian model averaging: A tutorial.Statistical Science 14,382\u2013417.","DOI":"10.1214\/ss\/1009212519"},{"key":"S026646660808016X_ref020","doi-asserted-by":"crossref","unstructured":"Mitchell, T. & J. 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Rao (2003)Detecting differentially expressed genes in microarrays using Bayesian model selection.Journal of the American Statistical Association 98,438\u2013455.","DOI":"10.1198\/016214503000224"},{"key":"S026646660808016X_ref026","doi-asserted-by":"crossref","unstructured":"Shao, J. (1996)Bootstrap model selection.Journal of the American Statistical Association 91,655\u2013665.","DOI":"10.1080\/01621459.1996.10476934"},{"key":"S026646660808016X_ref032","doi-asserted-by":"crossref","unstructured":"Zhang, P. (1993a)Model selection via multifold cross validation.Annals of Statistics 21,299\u2013313.","DOI":"10.1214\/aos\/1176349027"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S026646660808016X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T05:49:52Z","timestamp":1750484992000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S026646660808016X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,11,30]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,4]]}},"alternative-id":["S026646660808016X"],"URL":"https:\/\/doi.org\/10.1017\/s026646660808016x","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,11,30]]}}}