{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T16:12:33Z","timestamp":1792080753736,"version":"4.4.0"},"reference-count":177,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"vor","delay-in-days":2201,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1305154"],"award-info":[{"award-number":["DMS-1305154"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007812","name":"Royalty Research Fund at the University of Washington","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007812","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000879","name":"Alfred P. Sloan Fellowship","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000879","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,3,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We consider approximate Bayesian model choice for model selection problems that involve models whose Fisher information matrices may fail to be invertible along other competing submodels. Such singular models do not obey the regularity conditions underlying the derivation of Schwarz's Bayesian information criterion BIC and the penalty structure in BIC generally does not reflect the frequentist large sample behaviour of the marginal likelihood. Although large sample theory for the marginal likelihood of singular models has been developed recently, the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning. Guided by examples such as determining the number of components in mixture models, the number of factors in latent factor models or the rank in reduced rank regression, we propose a resolution to this paradox and give a practical extension of BIC for singular model selection problems.<\/jats:p>","DOI":"10.1111\/rssb.12187","type":"journal-article","created":{"date-parts":[[2017,2,8]],"date-time":"2017-02-08T04:26:39Z","timestamp":1486527999000},"page":"323-380","source":"Crossref","is-referenced-by-count":106,"title":["A Bayesian Information Criterion for Singular Models"],"prefix":"10.1093","volume":"79","author":[{"given":"Mathias","family":"Drton","sequence":"first","affiliation":[{"name":"University of Washington , Seattle , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martyn","family":"Plummer","sequence":"additional","affiliation":[{"name":"International Agency for Research on Cancer , Lyon , France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2017,2,7]]},"reference":[{"key":"2023021801103615400_rssb12187-cit-0001","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1177\/1471082X0100100404","article-title":"Likelihood and Bayesian analysis of mixtures","volume":"1","author":"Aitkin","year":"2001","journal-title":"Statist. Modllng"},{"key":"2023021801103615400_rssb12187-cit-0002","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEEE Trans. Autom. Control"},{"key":"2023021801103615400_rssb12187-cit-0003","doi-asserted-by":"crossref","first-page":"3099","DOI":"10.1214\/09-AOS689","article-title":"Identifiability of parameters in latent structure models with many observed variables","volume":"37","author":"Allman","year":"2009","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0004","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.laa.2013.10.046","article-title":"Tensors of nonnegative rank two","volume":"473","author":"Allman","year":"2015","journal-title":"Lin. Alg. Appl."},{"key":"2023021801103615400_rssb12187-cit-0005","first-page":"177","article-title":"Log canonical threshold of Vandermonde matrix type singularities and generalization error of a three-layered neural network in Bayesian estimation","volume":"52","author":"Aoyagi","year":"2009","journal-title":"Int. J. Pure Appl. Math."},{"key":"2023021801103615400_rssb12187-cit-0006","doi-asserted-by":"crossref","first-page":"2667","DOI":"10.1080\/03610920903094899","article-title":"A Bayesian learning coefficient of generalization error and Vandermonde matrix-type singularities","volume":"39","author":"Aoyagi","year":"2010","journal-title":"Communs Statist. Theory Meth."},{"key":"2023021801103615400_rssb12187-cit-0007","first-page":"1243","article-title":"Stochastic complexity and generalization error of a restricted Boltzmann machine in Bayesian estimation","volume":"11","author":"Aoyagi","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"2023021801103615400_rssb12187-cit-0008","doi-asserted-by":"crossref","first-page":"924","DOI":"10.1016\/j.neunet.2005.03.014","article-title":"Stochastic complexities of reduced rank regression in Bayesian estimation","volume":"18","author":"Aoyagi","year":"2005","journal-title":"Neurl Netwrks"},{"key":"2023021801103615400_rssb12187-cit-0009","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3940-6","volume-title":"Singularities of Differentiable Maps","author":"Arnol\u2019d","year":"1988"},{"key":"2023021801103615400_rssb12187-cit-0010","doi-asserted-by":"crossref","first-page":"775","DOI":"10.3150\/bj\/1161614946","article-title":"Asymptotic distribution and local power of the log-likelihood ratio test for mixtures: bounded and unbounded cases","volume":"12","author":"Aza\u00efs","year":"2006","journal-title":"Bernoulli"},{"key":"2023021801103615400_rssb12187-cit-0011","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1051\/ps:2008010","article-title":"The likelihood ratio test for general mixture models with or without structural parameter","volume":"13","author":"Aza\u00efs","year":"2009","journal-title":"ESAIM Probab. Statist."},{"key":"2023021801103615400_rssb12187-cit-0012","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/s11222-015-9561-x","article-title":"EM for mixtures","volume":"25","author":"Baudry","year":"2015","journal-title":"Statist. Comput."},{"key":"2023021801103615400_rssb12187-cit-0013","first-page":"83","volume-title":"Statistics and Probability: a Raghu Raj Bahadur Festschrift","author":"Bickel","year":"1993"},{"key":"2023021801103615400_rssb12187-cit-0014","volume-title":"Model Selection and Multimodel Inference","author":"Burnham","year":"2002"},{"key":"2023021801103615400_rssb12187-cit-0015","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.jpaa.2004.12.049","article-title":"Higher secant varieties of the Segre varieties P1\u00d7\u2026\u00d7P1","volume":"201","author":"Catalisano","year":"2005","journal-title":"J. Pure Appl. Alg."},{"key":"2023021801103615400_rssb12187-cit-0016","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1111\/j.1467-9469.2006.00532.x","article-title":"Semiparametric mixtures of generalized exponential families","volume":"34","author":"Charnigo","year":"2007","journal-title":"Scand. J. Statist."},{"key":"2023021801103615400_rssb12187-cit-0017","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1093\/biomet\/asn034","article-title":"Extended Bayesian information criterion for model selection with large model space","volume":"95","author":"Chen","year":"2008","journal-title":"Biometrika"},{"key":"2023021801103615400_rssb12187-cit-0018","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.jeconom.2012.01.022","article-title":"Cointegrating rank selection in models with time-varying variance","volume":"169","author":"Cheng","year":"2012","journal-title":"J. Econmetr."},{"key":"2023021801103615400_rssb12187-cit-0019","volume-title":"Model Selection and Model Averaging","author":"Claeskens","year":"2008"},{"key":"2023021801103615400_rssb12187-cit-0020","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1080\/01621459.1997.10474045","article-title":"Computing Bayes factors by combining simulation and asymptotic approximations","volume":"92","author":"DiCiccio","year":"1997","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-0021","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1214\/07-AOS571","article-title":"Likelihood ratio tests and singularities","volume":"37","author":"Drton","year":"2009","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0022","doi-asserted-by":"crossref","DOI":"10.3150\/15-BEJ775","article-title":"Marginal likelihood and model selection for Gaussian latent tree and forest models","author":"Drton","year":"2017","journal-title":"Bernoulli"},{"key":"2023021801103615400_rssb12187-cit-0023","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/s00440-006-0033-2","article-title":"Algebraic factor analysis: tetrads, pentads and beyond","volume":"138","author":"Drton","year":"2007","journal-title":"Probab. Theory Reltd Flds"},{"key":"2023021801103615400_rssb12187-cit-0024","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-7643-8905-5","volume-title":"Lectures on Algebraic Statistics","author":"Drton","year":"2009"},{"key":"2023021801103615400_rssb12187-cit-0025","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1111\/j.1467-9868.2011.01025.x","article-title":"Catching up faster by switching sooner: a predictive approach to adaptive estimation with an application to the AIC\u2013BIC dilemma (with discussion)","volume":"74","author":"van Erven","year":"2012","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-0026","first-page":"2020","article-title":"Extended Bayesian information criteria for Gaussian graphical models","volume":"23","author":"Foygel","year":"2010","journal-title":"Adv. Neurl Inf. Process. Syst."},{"key":"2023021801103615400_rssb12187-cit-0027","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s00357-007-0004-5","article-title":"Bayesian regularization for normal mixture estimation and model-based clustering","volume":"24","author":"Fraley","year":"2007","journal-title":"J. Classificn"},{"key":"2023021801103615400_rssb12187-cit-0028","volume-title":"MCLUST version 4 for R: normal mixture modeling for model-based clustering, classification, and density estimation","author":"Fraley","year":"2012"},{"key":"2023021801103615400_rssb12187-cit-0029","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1111\/j.1467-9868.2007.00650.x","article-title":"Marginal likelihood estimation via power posteriors","volume":"70","author":"Friel","year":"2008","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-0030","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1111\/j.1467-9574.2011.00515.x","article-title":"Estimating the evidence\u2014a review","volume":"66","author":"Friel","year":"2012","journal-title":"Statist. Neerland."},{"key":"2023021801103615400_rssb12187-cit-0031","volume-title":"Finite Mixture and Markov Switching Models","author":"Fr\u00fchwirth-Schnatter","year":"2006"},{"key":"2023021801103615400_rssb12187-cit-0032","first-page":"1123","article-title":"Tuning parameter selection for penalized likelihood estimation of Gaussian graphical model","volume":"22","author":"Gao","year":"2012","journal-title":"Statist. Sin."},{"key":"2023021801103615400_rssb12187-cit-0033","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1109\/TIT.2012.2221122","article-title":"Consistent order estimation and minimal penalties","volume":"59","author":"Gassiat","year":"2013","journal-title":"IEEE Trans. Inform. Theory"},{"key":"2023021801103615400_rssb12187-cit-0034","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1090\/S0002-9947-2013-06041-2","article-title":"The local geometry of finite mixtures","volume":"366","author":"Gassiat","year":"2014","journal-title":"Trans. Am. Math. Soc."},{"key":"2023021801103615400_rssb12187-cit-0035","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1214\/aos\/1009210550","article-title":"Stratified exponential families: graphical models and model selection","volume":"29","author":"Geiger","year":"2001","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0036","first-page":"807","volume-title":"Proc. Berkeley Conf. in Honor of Jerzy Neyman and Jack Kiefer","author":"Hartigan","year":"1985"},{"key":"2023021801103615400_rssb12187-cit-0037","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-84858-7","volume-title":"The Elements of Statistical Learning","author":"Hastie","year":"2009"},{"key":"2023021801103615400_rssb12187-cit-0038","first-page":"45","article-title":"Size of the error in the choice of a model to fit data from an exponential family","volume":"51","author":"Haughton","year":"1989","journal-title":"Sankhya"},{"key":"2023021801103615400_rssb12187-cit-0039","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1214\/aos\/1176350709","article-title":"On the choice of a model to fit data from an exponential family","volume":"16","author":"Haughton","year":"1988","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0040","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1214\/ss\/1009212519","article-title":"Bayesian model averaging: a tutorial (with comments)","volume":"14","author":"Hoeting","year":"1999","journal-title":"Statist. Sci."},{"key":"2023021801103615400_rssb12187-cit-0041","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1080\/01621459.1995.10476592","article-title":"A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-0042","first-page":"49","article-title":"Consistent estimation of the order of mixture models","volume":"62","author":"Keribin","year":"2000","journal-title":"Sankhya"},{"key":"2023021801103615400_rssb12187-cit-0043","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-71887-3","volume-title":"Information Criteria and Statistical Modeling","author":"Konishi","year":"2008"},{"key":"2023021801103615400_rssb12187-cit-0044","volume-title":"Importance sampling schemes for evidence approximation in mixture models","author":"Lee","year":"2013"},{"key":"2023021801103615400_rssb12187-cit-0045","volume-title":"Asymptotic approximation of marginal likelihood integrals","author":"Lin","year":"2011"},{"key":"2023021801103615400_rssb12187-cit-0046","first-page":"1","article-title":"poLCA: an R package for polytomous variable latent class analysis","volume":"42","author":"Linzer","year":"2011","journal-title":"J. Statist. Softwr."},{"key":"2023021801103615400_rssb12187-cit-0047","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1214\/aos\/1056562463","article-title":"Asymptotics for likelihood ratio tests under loss of identifiability","volume":"31","author":"Liu","year":"2003","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0048","first-page":"41","article-title":"Bayesian model assessment in factor analysis","volume":"14","author":"Lopes","year":"2004","journal-title":"Statist. Sin."},{"key":"2023021801103615400_rssb12187-cit-0049","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1016\/S0169-7161(05)25016-2","volume-title":"Bayesian Thinking: Modeling and Computation","author":"Marin","year":"2005"},{"key":"2023021801103615400_rssb12187-cit-0050","author":"McLachlan","year":"2000"},{"key":"2023021801103615400_rssb12187-cit-0051","volume-title":"Erroneous results in \u2018Marginal likelihood from the Gibbs output\u2019","author":"Neal","year":"1999"},{"key":"2023021801103615400_rssb12187-cit-0052","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1214\/aos\/1176346522","article-title":"Asymptotic properties of criteria for selection of variables in multiple regression","volume":"12","author":"Nishii","year":"1984","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0053","volume-title":"Bayesian finite mixtures: a note on prior specification and posterior computation","author":"Nobile","year":"2005"},{"key":"2023021801103615400_rssb12187-cit-0054","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1080\/10705510701575396","article-title":"Deciding on the number of classes in latent class analysis and growth mixture modeling: a Monte Carlo simulation study","volume":"14","author":"Nylund","year":"2007","journal-title":"Struct. Equn Modlng"},{"key":"2023021801103615400_rssb12187-cit-0055","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1214\/aos\/1176342472","article-title":"Distinctness of the eigenvalues of a quadratic form in a multivariate sample","volume":"1","author":"Okamoto","year":"1973","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0056","first-page":"1889","volume-title":"Repulsive mixtures. In Advances in Neural Information Processing Systems","author":"Petralia","year":"2012"},{"key":"2023021801103615400_rssb12187-cit-0057","doi-asserted-by":"crossref","first-page":"111","DOI":"10.2307\/271063","article-title":"Bayesian model selection in social research","volume":"25","author":"Raftery","year":"1995","journal-title":"Sociol. Methodol."},{"key":"2023021801103615400_rssb12187-cit-0058","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2853-8","volume-title":"Multivariate Reduced-rank Regression","author":"Reinsel","year":"1998"},{"key":"2023021801103615400_rssb12187-cit-0059","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1111\/1467-9868.00095","article-title":"On Bayesian analysis of mixtures with an unknown number of components (with discussion)","volume":"59","author":"Richardson","year":"1997","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-0060","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1080\/01621459.1997.10474044","article-title":"Practical Bayesian density estimation using mixtures of normals","volume":"92","author":"Roeder","year":"1997","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-0061","doi-asserted-by":"crossref","first-page":"243","DOI":"10.2307\/3318576","article-title":"Likelihood-based inference with singular information matrix","volume":"6","author":"Rotnitzky","year":"2000","journal-title":"Bernoulli"},{"key":"2023021801103615400_rssb12187-cit-0062","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1111\/j.1467-9868.2011.00781.x","article-title":"Asymptotic behaviour of the posterior distribution in overfitted mixture models","volume":"73","author":"Rousseau","year":"2011","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-0063","first-page":"1","article-title":"Asymptotic model selection for naive Bayesian networks","volume":"6","author":"Rusakov","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"2023021801103615400_rssb12187-cit-0064","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0065","doi-asserted-by":"crossref","first-page":"2587","DOI":"10.1214\/10-AOS792","article-title":"Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem","volume":"38","author":"Scott","year":"2010","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-0066","first-page":"113","volume-title":"Frontiers of Statistical Decision Making and Bayesian Analysis","author":"Steele","year":"2010"},{"key":"2023021801103615400_rssb12187-cit-0067","doi-asserted-by":"crossref","first-page":"1265","DOI":"10.1214\/aoms\/1177703862","article-title":"Identifiability of finite mixtures","volume":"34","author":"Teicher","year":"1963","journal-title":"Ann. Math. Statist."},{"key":"2023021801103615400_rssb12187-cit-0068","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1080\/01621459.1986.10478240","article-title":"Accurate approximations for posterior moments and marginal densities","volume":"81","author":"Tierney","year":"1986","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-0069","volume-title":"Statistical Analysis of Finite Mixture Distributions","author":"Titterington","year":"1985"},{"key":"2023021801103615400_rssb12187-cit-0070","first-page":"1","article-title":"Asymptotic behavior of exponential integrals in the complex domain","volume":"13","author":"Vasil\u2019ev","year":"1979","journal-title":"Funkt. Anal. Prilzhn."},{"key":"2023021801103615400_rssb12187-cit-0071","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1006\/jmps.1999.1278","article-title":"Bayesian model selection and model averaging","volume":"44","author":"Wasserman","year":"2000","journal-title":"J. Math. Psychol."},{"key":"2023021801103615400_rssb12187-cit-0072","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1162\/089976601300014402","article-title":"Algebraic analysis for nonidentifiable learning machines","volume":"13","author":"Watanabe","year":"2001","journal-title":"Neurl Comput."},{"key":"2023021801103615400_rssb12187-cit-0073","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511800474","volume-title":"Algebraic Geometry and Statistical Learning Theory","author":"Watanabe","year":"2009"},{"key":"2023021801103615400_rssb12187-cit-0074","first-page":"867","article-title":"A widely applicable Bayesian information criterion","volume":"14","author":"Watanabe","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"2023021801103615400_rssb12187-cit-0075","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1162\/089976603765202640","article-title":"Learning coefficients of layered models when the true distribution mismatches the singularities","volume":"15","author":"Watanabe","year":"2003","journal-title":"Neurl Comput."},{"key":"2023021801103615400_rssb12187-cit-0076","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.tcs.2007.07.039","article-title":"Stochastic complexity for mixture of exponential families in generalized variational Bayes","volume":"387","author":"Watanabe","year":"2007","journal-title":"Theoret. Comput. Sci."},{"key":"2023021801103615400_rssb12187-cit-0100","volume-title":"sBIC: computing the singular BIC for multiple models","author":"Weihs","year":"2016"},{"key":"2023021801103615400_rssb12187-cit-0077","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1111\/j.1467-9574.2012.00530.x","article-title":"All models are wrong ...\u2019: an introduction to model uncertainty","volume":"66","author":"Wit","year":"2012","journal-title":"Statist. Neerland."},{"key":"2023021801103615400_rssb12187-cit-0078","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719260","volume-title":"Asymptotic Approximations of Integrals","author":"Wong","year":"2001"},{"key":"2023021801103615400_rssb12187-cit-0079","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1016\/S0893-6080(03)00005-4","article-title":"Singularities in mixture models and upper bounds of stochastic complexity","volume":"16","author":"Yamazaki","year":"2003","journal-title":"Neurl Netwrks"},{"key":"2023021801103615400_rssb12187-cit-0080","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1007\/978-3-540-30215-5_27","volume-title":"Algorithmic Learning Theory","author":"Yamazaki","year":"2004"},{"key":"2023021801103615400_rssb12187-cit-0081","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.neucom.2005.02.014","article-title":"Algebraic geometry and stochastic complexity of hidden Markov models","volume":"69","author":"Yamazaki","year":"2005","journal-title":"Neurocomputing"},{"key":"2023021801103615400_rssb12187-cit-0082","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1093\/biomet\/92.4.937","article-title":"Can the strengths of AIC and BIC be shared?: a conflict between model identification and regression estimation","volume":"92","author":"Yang","year":"2005","journal-title":"Biometrika"},{"key":"2023021801103615400_rssb12187-cit-0083","first-page":"3283","article-title":"An asymptotic behaviour of the marginal likelihood for general Markov models","volume":"12","author":"Zwiernik","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"2023021801103615400_rssb12187-cit-0084","doi-asserted-by":"crossref","first-page":"290","DOI":"10.3150\/10-BEJ338","article-title":"Tree cumulants and the geometry of binary tree models","volume":"18","author":"Zwiernik","year":"2012","journal-title":"Bernoulli"},{"key":"2023021801103615400_rssb12187-cit-1001","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1111\/j.2517-6161.1985.tb01331.x","article-title":"Estimation and hypothesis testing in finite mixture models","volume":"47","author":"Aitkin","year":"1985","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1002","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/BF02532251","article-title":"Fitting autoregressive models for prediction","volume":"21","author":"Akaike","year":"1969","journal-title":"Ann. Inst. Statist. Math."},{"key":"2023021801103615400_rssb12187-cit-1003","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/BF02506337","article-title":"Statistical predictor identification","volume":"22","author":"Akaike","year":"1970","journal-title":"Ann. Inst. Statist. Math."},{"key":"2023021801103615400_rssb12187-cit-1004","first-page":"267","article-title":"Information theory and an extension of the maximum likelihood principle","author":"Akaike","year":"1973","journal-title":"Proc. 2nd Int. Symp. Information Theory"},{"key":"2023021801103615400_rssb12187-cit-1005","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-9473(01)00046-9","article-title":"A mixture model approach for the analysis of microarray gene expression data","volume":"39","author":"Allison","year":"2002","journal-title":"Computnl Statist. Data Anal."},{"key":"2023021801103615400_rssb12187-cit-1006","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/s11222-014-9521-x","article-title":"Noisy Monte Carlo: convergence of Markov chains with approximate transition kernels","volume":"26","author":"Alquier","year":"2016","journal-title":"Statist. Comput."},{"key":"2023021801103615400_rssb12187-cit-1007","volume-title":"Stepwise searching for feature variables in high-dimensional linear regression","author":"An","year":"2008"},{"key":"2023021801103615400_rssb12187-cit-1008","first-page":"177","article-title":"Log canonical threshold of Vandermonde matrix type singularities and generalization error of a three-layered neural network in Bayesian estimation","volume":"52","author":"Aoyagi","year":"2009","journal-title":"Int. J. Pure Appl. Math."},{"key":"2023021801103615400_rssb12187-cit-1009","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1162\/NECO_a_00271","article-title":"Learning coefficient of generalization error in Bayesian estimation and Vandermonde matrix-type singularity","volume":"24","author":"Aoyagi","year":"2012","journal-title":"Neurl Computn"},{"key":"2023021801103615400_rssb12187-cit-1010","first-page":"171","article-title":"A class of distributions which includes the normal ones","volume":"12","author":"Azzalini","year":"1985","journal-title":"Scand. J. Statist."},{"key":"2023021801103615400_rssb12187-cit-1011","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/978-3-319-27099-9_2","volume-title":"Statistical Analysis for High-dimensional Data: the Abel Symposium 2014","author":"Barber","year":"2016"},{"key":"2023021801103615400_rssb12187-cit-1012","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1198\/jcgs.2010.08111","article-title":"Combining mixture components for clustering","volume":"9","author":"Baudry","year":"2010","journal-title":"J. Computnl Graph. Statist."},{"key":"2023021801103615400_rssb12187-cit-1013","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316870","volume-title":"Bayesian Theory","author":"Bernardo","year":"1994"},{"key":"2023021801103615400_rssb12187-cit-1014","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/BF00050857","article-title":"Asymptotically efficient autoregressive model selection for multistep prediction","volume":"48","author":"Bhansali","year":"1996","journal-title":"Ann. Inst Statist. Math."},{"key":"2023021801103615400_rssb12187-cit-1015","first-page":"547","article-title":"Some properties of the order of an autoregressive model selected by a generalization of Akaike's FPE criterion","volume":"64","author":"Bhansali","year":"1977","journal-title":"Biometrika"},{"key":"2023021801103615400_rssb12187-cit-1016","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.csda.2012.12.008","article-title":"Model-based clustering of high-dimensional data: a review","volume":"71","author":"Bouveyron","year":"2014","journal-title":"Computnl Statist. Data Anal."},{"key":"2023021801103615400_rssb12187-cit-1017","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1214\/15-AOAS861","article-title":"The discriminative functional mixture model for a comparative analysis of bike sharing systems","volume":"9","author":"Bouveyron","year":"2015","journal-title":"Ann. Appl. Statist."},{"key":"2023021801103615400_rssb12187-cit-1018","volume-title":"Bayesian variable selection for globally sparse probabilistic PCA","author":"Bouveyron","year":"2016"},{"key":"2023021801103615400_rssb12187-cit-1019","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1016\/j.jspi.2004.06.062","article-title":"A note on reference priors for the scalar skew-normal distribution","volume":"136","author":"Brunero","year":"2006","journal-title":"J. Statist. Planng Inf."},{"key":"2023021801103615400_rssb12187-cit-1020","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1214\/13-STS465","article-title":"Recursive pathways to marginal likelihood estimation with prior sensitivity analysis","volume":"29","author":"Cameron","year":"2014","journal-title":"Statist. Sci."},{"key":"2023021801103615400_rssb12187-cit-1021","volume-title":"Bayes and Empirical Bayes Methods for Data Analysis","author":"Carlin","year":"2001"},{"key":"2023021801103615400_rssb12187-cit-1022","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1214\/009053607000000857","article-title":"Bounds for Bayesian order identification with application to mixtures","volume":"36","author":"Chambaz","year":"2008","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-1023","article-title":"Supplementary material for \u201cAsymptotic relationships between the D-test and likelihood ratio-type tests for homogeneity\u201d","volume":"20","author":"Charnigo","year":"2010","journal-title":"Statist. Sin."},{"key":"2023021801103615400_rssb12187-cit-1024","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1093\/biomet\/asn034","article-title":"Extended Bayesian information criteria for model selection with large model spaces","volume":"95","author":"Chen","year":"2008","journal-title":"Biometrika"},{"key":"2023021801103615400_rssb12187-cit-1025","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1111\/1467-9868.00273","article-title":"A modified likelihood ratio test for homogeneity in finite mixture models","volume":"63","author":"Chen","year":"2001","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1026","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1214\/aoms\/1177728725","article-title":"On the distribution of the likelihood ratio","volume":"25","author":"Chernoff","year":"1954","journal-title":"Ann. Math. Statist."},{"key":"2023021801103615400_rssb12187-cit-1027","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1093\/biomet\/89.3.539","article-title":"A sequential particle filter method for static models","volume":"89","author":"Chopin","year":"2002","journal-title":"Biometrika"},{"key":"2023021801103615400_rssb12187-cit-1028","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1023\/A:1022415228062","article-title":"Likelihood ratio statistic for exponential mixtures","volume":"54","author":"Ciuperca","year":"2002","journal-title":"Ann. Inst. Statist. Math."},{"key":"2023021801103615400_rssb12187-cit-1029","doi-asserted-by":"crossref","first-page":"900","DOI":"10.1198\/016214503000000819","article-title":"The focused information criterion","volume":"98","author":"Claeskens","year":"2003","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-1030","first-page":"842","article-title":"Inferences in contaminated regression and density models","volume":"69","author":"Dai","year":"2008","journal-title":"Sankhya"},{"key":"2023021801103615400_rssb12187-cit-1031","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1080\/01621459.1998.10474110","article-title":"Detecting features in spatial point processes with clutter via model-based clustering","volume":"93","author":"Dasgupta","year":"1998","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-1032","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1007\/s10463-009-0258-9","article-title":"Latent class analysis variable selection","volume":"62","author":"Dean","year":"2010","journal-title":"Ann. Inst. Statist. Math."},{"key":"2023021801103615400_rssb12187-cit-1033","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1111\/j.1467-9868.2006.00553.x","article-title":"Sequential Monte Carlo samplers","volume":"68","author":"Del Moral","year":"2006","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1034","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1093\/acprof:oso\/9780199695607.003.0020","volume-title":"Bayesian Theory and Applications","author":"Draper","year":"2013"},{"key":"2023021801103615400_rssb12187-cit-1035","doi-asserted-by":"crossref","DOI":"10.3150\/15-BEJ775","article-title":"Marginal likelihood and model selection for Gaussian latent tree and forest models","author":"Drton","year":"2017","journal-title":"Bernoulli"},{"key":"2023021801103615400_rssb12187-cit-1036","volume-title":"Importance nested sampling and MULTINEST algorithm","author":"Feroz","year":"2014"},{"key":"2023021801103615400_rssb12187-cit-1037","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1093\/comjnl\/41.8.578","article-title":"How many clusters?: which cluster method?; answers via model-based cluster analysis","volume":"41","author":"Fraley","year":"1998","journal-title":"Comput. J."},{"key":"2023021801103615400_rssb12187-cit-1038","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1198\/016214502760047131","article-title":"Model-based clustering, discriminant analysis, and density estimation","volume":"97","author":"Fraley","year":"2002","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-1039","article-title":"Investigation of the widely applicable Bayesian information criterion","author":"Friel","year":"2016","journal-title":"Statist. Comput."},{"key":"2023021801103615400_rssb12187-cit-1040","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1111\/j.1467-9868.2007.00650.x","article-title":"Marginal likelihood estimation via power posteriors","volume":"70","author":"Friel","year":"2008","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1041","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1109\/TIT.2012.2221122","article-title":"Consistent order estimation and minimal penalties","volume":"59","author":"Gassiat","year":"2013","journal-title":"IEEE Trans. Inform. Theory"},{"key":"2023021801103615400_rssb12187-cit-1042","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/biomet\/85.1.1","article-title":"Model choice: a minimum posterior predictive loss approach","volume":"85","author":"Gelfand","year":"1998","journal-title":"Biometrika"},{"key":"2023021801103615400_rssb12187-cit-1043","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1214\/ss\/1028905934","article-title":"Simulating normalizing constants: from importance sampling to bridge sampling to path sampling","volume":"13","author":"Gelman","year":"1998","journal-title":"Statist. Sci."},{"key":"2023021801103615400_rssb12187-cit-1044","volume-title":"Estimating normalizing constants and reweighting mixtures","author":"Geyer","year":"1994"},{"key":"2023021801103615400_rssb12187-cit-1045","first-page":"807","volume-title":"Proc. Berkeley Conf. in Honor of Jerzy Neyman and Jack Kiefer","author":"Hartigan","year":"1985"},{"key":"2023021801103615400_rssb12187-cit-1046","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pone.0131739","article-title":"Overfitting Bayesian mixture models with an unknown number of components","author":"van Havre","year":"2015","journal-title":"PLOS ONE"},{"key":"2023021801103615400_rssb12187-cit-1047","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s11634-010-0058-3","article-title":"Methods for merging Gaussian mixture components","volume":"4","author":"Hennig","year":"2010","journal-title":"Adv. Data Anal. Classificn"},{"key":"2023021801103615400_rssb12187-cit-1048","first-page":"109","volume-title":"Classification as a Tool for Research, Studies in Classification, Data Analysis, and Knowledge Organization","author":"Hennig","year":"2010"},{"key":"2023021801103615400_rssb12187-cit-1049","first-page":"382","article-title":"Bayesian model averaging: a tutorial","author":"Hoeting","year":"1999","journal-title":"Statist. Sci."},{"key":"2023021801103615400_rssb12187-cit-1050","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139025751","volume-title":"Causal Inference for Statistics, Social, and Biomedical Sciences: an Introduction","author":"Imbens","year":"2015"},{"key":"2023021801103615400_rssb12187-cit-1051","doi-asserted-by":"crossref","first-page":"928","DOI":"10.1080\/01621459.1995.10476592","article-title":"A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-1052","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1111\/1467-9868.00404","article-title":"A theory of statistical models for Monte Carlo integration","volume":"65","author":"Kong","year":"2003","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1053","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.aca.2008.10.069","article-title":"Determination of glucose and ethanol in bioethanol production by near infrared spectroscopy and chemometrics","volume":"642","author":"Liebmann","year":"2009","journal-title":"Anal. Clim. Acta"},{"key":"2023021801103615400_rssb12187-cit-1054","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.jmva.2008.04.010","article-title":"Maximum likelihood estimation for multivariate skew normal mixture models","volume":"100","author":"Lin","year":"2009","journal-title":"J. Multiv. Anal."},{"key":"2023021801103615400_rssb12187-cit-1055","doi-asserted-by":"crossref","DOI":"10.5705\/ss.202015.0212","article-title":"Estimation under model uncertainty","author":"Longford","year":"2017","journal-title":"Statist. Sin."},{"key":"2023021801103615400_rssb12187-cit-1056","first-page":"41","article-title":"Bayesian model assessment in factor analysis","volume":"14","author":"Lopes","year":"2004","journal-title":"Statist. Sin."},{"key":"2023021801103615400_rssb12187-cit-1057","volume-title":"Concentration Inequalities and Model Selection","author":"Massart","year":"2007"},{"key":"2023021801103615400_rssb12187-cit-1058","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1080\/10618600.2014.978007","article-title":"Merging mixture components for clustering through pairwise overlap","volume":"26","author":"Melnykov","year":"2016","journal-title":"J. Computnl Graph. Statist."},{"key":"2023021801103615400_rssb12187-cit-1059","doi-asserted-by":"crossref","first-page":"1937","DOI":"10.1214\/09-AOS754","article-title":"Consistency of objective Bayes factors as the model dimension grows","volume":"38","author":"Moreno","year":"2010","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-1060","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1080\/10705510701575396","article-title":"Deciding on the number of classes in latent class analysis and growth mixture modelling: a Monte Carlo simulation study","volume":"14","author":"Nylund","year":"2007","journal-title":"Struct. Equn Modlng"},{"key":"2023021801103615400_rssb12187-cit-1061","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1080\/01621459.2015.1021006","article-title":"The controlled thermodynamic integral for Bayesian model evidence evaluation","volume":"111","author":"Oates","year":"2016","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-1062","first-page":"446","volume-title":"Proc. 16th A. Conf. Uncertainty in Artifical Intelligence","author":"Ortiz","year":"2000"},{"key":"2023021801103615400_rssb12187-cit-1063","first-page":"1889","volume-title":"Advances in Neural Information Processing Systems","author":"Petralia","year":"2012"},{"key":"2023021801103615400_rssb12187-cit-1064","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1111\/j.2517-6161.1987.tb01700.x","article-title":"A note on the Aitkin-Rubin approach to hypothesis testing in mixture models","volume":"49","author":"Quinn","year":"1987","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1065","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1177\/0049124199027003005","article-title":"Bayes factors and BIC\u2014Comment on \u201cA critique of the Bayesian information criterion for model selection\u201d","volume":"27","author":"Raftery","year":"1999","journal-title":"Sociol. Meth. Res."},{"key":"2023021801103615400_rssb12187-cit-1066","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1111\/1467-9868.00095","article-title":"On Bayesian analysis of mixtures with an unknown number of components (with discussion)","volume":"59","author":"Richardson","year":"1997","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1067","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1080\/01621459.1990.10474918","article-title":"Density estimation with confidence sets exemplified by superclusters and voids in the galaxies","volume":"85","author":"Roeder","year":"1990","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-1068","doi-asserted-by":"crossref","first-page":"894","DOI":"10.1080\/01621459.1997.10474044","article-title":"Practical Bayesian density estimation using mixtures of normals","volume":"92","author":"Roeder","year":"1997","journal-title":"J. Am. Statist. Ass."},{"key":"2023021801103615400_rssb12187-cit-1069","volume-title":"Bayesian Statistics 8","author":"Rousseau","year":"2007"},{"key":"2023021801103615400_rssb12187-cit-1070","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1111\/j.1467-9868.2011.00781.x","article-title":"Asymptotic behaviour of the posterior distribution in overfitted mixture models","volume":"73","author":"Rousseau","year":"2011","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1071","volume-title":"The Cross-entropy Method: a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning","author":"Rubinstein","year":"2013"},{"key":"2023021801103615400_rssb12187-cit-1072","doi-asserted-by":"crossref","first-page":"4259","DOI":"10.1016\/j.jspi.2005.08.043","article-title":"Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions","volume":"136","author":"Sartori","year":"2006","journal-title":"J. Statist. Planng Inf."},{"key":"2023021801103615400_rssb12187-cit-1073","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s11222-011-9299-z","article-title":"Sequential Monte Carlo on large binary sampling spaces","volume":"23","author":"Sch\u00e4fer","year":"2013","journal-title":"Statist. Comput."},{"key":"2023021801103615400_rssb12187-cit-1074","first-page":"14429","volume-title":"Proc. Natn. Acad. Sci. USA","author":"Scheetz","year":"2006"},{"key":"2023021801103615400_rssb12187-cit-1075","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-1076","doi-asserted-by":"crossref","first-page":"289","DOI":"10.32614\/RJ-2016-021","article-title":"mclust 5: clustering classification and density estimation using Gaussian finite mixture models","volume":"8","author":"Scrucca","year":"2016","journal-title":"R J."},{"key":"2023021801103615400_rssb12187-cit-1077","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1093\/biomet\/63.1.117","article-title":"Selection of the order of an autoregressive model by Akaike's information criterion","volume":"63","author":"Shibata","year":"1976","journal-title":"Biometrika"},{"key":"2023021801103615400_rssb12187-cit-100501","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1214\/aos\/1176344897","article-title":"Asymptotically efficient selection of the order for estimating the parameters of a linear process","volume":"8","author":"Shibata","year":"1980","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-1078","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1111\/j.2517-6161.1995.tb02060.x","article-title":"Laplace approximation of high dimensional integrals","volume":"57","author":"Shun","year":"1995","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1079","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1111\/1467-9868.00353","article-title":"Bayesian measures of model complexity and fit (with discussion)","volume":"64","author":"Spiegelhalter","year":"2002","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1080","first-page":"113","volume-title":"Frontiers of Statistical Decision Making and Bayesian Analysis","author":"Steele","year":"2010"},{"key":"2023021801103615400_rssb12187-cit-1081","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s001840200206","article-title":"Distribution of exact tests in the exponential family","volume":"57","author":"Stehl\u00edk","year":"2003","journal-title":"Metrika"},{"key":"2023021801103615400_rssb12187-cit-1082","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1080\/03610918.2010.551011","article-title":"Exact likelihood ratio testing for homogeneity of exponential distribution","volume":"40","author":"Stehl\u00edk","year":"2011","journal-title":"Communs Statist. Simuln Computn"},{"key":"2023021801103615400_rssb12187-cit-1083","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1111\/1467-9868.00196","article-title":"Probabilistic principal component analysis","volume":"61","author":"Tipping","year":"1999","journal-title":"J. R. Statist. Soc."},{"key":"2023021801103615400_rssb12187-cit-1084","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1214\/aos\/1176346585","article-title":"Empirical distributions in selection bias models","volume":"13","author":"Vardi","year":"1985","journal-title":"Ann. Statist."},{"key":"2023021801103615400_rssb12187-cit-1085","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511800474","volume-title":"Algebraic Geometry and Statistical Learning Theory","author":"Watanabe","year":"2009"},{"key":"2023021801103615400_rssb12187-cit-1086","first-page":"3571","article-title":"Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory","volume":"11","author":"Watanabe","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"2023021801103615400_rssb12187-cit-1087","first-page":"867","article-title":"A widely applicable Bayesian information criterion","volume":"14","author":"Watanabe","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"2023021801103615400_rssb12187-cit-1088","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1007\/s11222-014-9542-5","article-title":"Bayesian variable selection for latent class analysis using a collapsed Gibbs sampler","volume":"26","author":"White","year":"2016","journal-title":"Statist. Comput."},{"key":"2023021801103615400_rssb12187-cit-1089","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1214\/aoms\/1177732360","article-title":"The large-sample distribution of the likelihood ratio for testing composite hypotheses","volume":"9","author":"Wilks","year":"1938","journal-title":"Ann. Math. Statist."},{"key":"2023021801103615400_rssb12187-cit-1090","doi-asserted-by":"crossref","first-page":"1814","DOI":"10.1214\/13-AOAS657","article-title":"A decision-theoretic approach for segmental classification","volume":"7","author":"Yau","year":"2013","journal-title":"Ann. Appl. Statist."},{"key":"2023021801103615400_rssb12187-cit-1091","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1080\/10618600.2015.1060885","article-title":"Towards automatic model comparison: an adaptive sequential Monte Carlo approach","volume":"25","author":"Zhou","year":"2016","journal-title":"J. Computnl Graph. Statist."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12187","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/rssb.12187","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/rssb.12187","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/am-pdf\/10.1111\/rssb.12187","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/2\/323\/49232168\/jrsssb_79_2_323.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/79\/2\/323\/49232168\/jrsssb_79_2_323.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,15]],"date-time":"2025-06-15T03:13:44Z","timestamp":1749957224000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/79\/2\/323\/7041940"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,7]]},"references-count":177,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,2,7]]},"published-print":{"date-parts":[[2017,3,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12187","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2017,3]]},"published":{"date-parts":[[2017,2,7]]}}}