{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T15:17:33Z","timestamp":1790781453610,"version":"4.1.0"},"reference-count":45,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2019,8,25]],"date-time":"2019-08-25T00:00:00Z","timestamp":1566691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"European Research Council"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,12,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>Regression models for dichotomous data are ubiquitous in statistics. Besides being useful for inference on binary responses, these methods serve as building blocks in more complex formulations, such as density regression, nonparametric classification and graphical models. Within the Bayesian framework, inference proceeds by updating the priors for the coefficients, typically taken to be Gaussians, with the likelihood induced by probit or logit regressions for the responses. In this updating, the apparent absence of a tractable posterior has motivated a variety of computational methods, including Markov chain Monte Carlo routines and algorithms that approximate the posterior. Despite being implemented routinely, Markov chain Monte Carlo strategies have mixing or time-inefficiency issues in large-$p$ and small-$n$ studies, whereas approximate routines fail to capture the skewness typically observed in the posterior. In this article it is proved that the posterior distribution for the probit coefficients has a unified skew-normal kernel under Gaussian priors. This result allows efficient Bayesian inference for a wide class of applications, especially in large-$p$ and small-to-moderate-$n$ settings where state-of-the-art computational methods face notable challenges. These advances are illustrated in a genetic study, and further motivate the development of a wider class of conjugate priors for probit models, along with methods for obtaining independent and identically distributed samples from the unified skew-normal posterior.<\/jats:p>","DOI":"10.1093\/biomet\/asz034","type":"journal-article","created":{"date-parts":[[2019,6,12]],"date-time":"2019-06-12T19:12:19Z","timestamp":1560366739000},"page":"765-779","source":"Crossref","is-referenced-by-count":48,"title":["Conjugate Bayes for probit regression via unified skew-normal distributions"],"prefix":"10.1093","volume":"106","author":[{"given":"Daniele","family":"Durante","sequence":"first","affiliation":[{"name":"Department of Decision Sciences, Bocconi University, Via R\u00f6ntgen 1, 20136 Milan, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2019,8,25]]},"reference":[{"key":"2019111117080417500_B1","author":"Agresti","year":"2013","journal-title":"Categorical Data Analysis"},{"key":"2019111117080417500_B2","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1080\/01621459.1993.10476321","article-title":"Bayesian analysis of binary and polychotomous response data","volume":"88","author":"Albert","year":"1993","journal-title":"J. Am. Statist. Assoc."},{"key":"2019111117080417500_B3","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1111\/j.1467-9469.2006.00503.x","article-title":"On the unification of families of skew-normal distributions","volume":"33","author":"Arellano-Valle","year":"2006","journal-title":"Scand. J. Statist."},{"key":"2019111117080417500_B4","first-page":"23","article-title":"Hidden truncation models","volume":"62","author":"Arnold","year":"2000","journal-title":"Sankhy\u0101"},{"key":"2019111117080417500_B5","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/BF02595728","article-title":"Skewed multivariate models related to hidden truncation and\/or selective reporting","volume":"11","author":"Arnold","year":"2002","journal-title":"Test"},{"key":"2019111117080417500_B6","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":"2019111117080417500_B7","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF03263544","article-title":"A prospective combination of phase II and phase III in drug development","volume":"68","author":"Azzalini","year":"2010","journal-title":"Metron"},{"key":"2019111117080417500_B8","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1111\/1467-9868.00194","article-title":"Statistical applications of the multivariate skew normal distribution","volume":"61","author":"Azzalini","year":"1999","journal-title":"J. R. Statist. Soc. B"},{"key":"2019111117080417500_B9","author":"Azzalini","year":"2014","journal-title":"The Skew-Normal and Related Families"},{"key":"2019111117080417500_B10","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1093\/biomet\/83.4.715","article-title":"The multivariate skew-normal distribution","volume":"83","author":"Azzalini","year":"1996","journal-title":"Biometrika"},{"key":"2019111117080417500_B11","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1214\/06-BA128","article-title":"A skew item response model","volume":"1","author":"Baz\u00e1n","year":"2006","journal-title":"Bayesian Anal."},{"key":"2019111117080417500_B12","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1111\/rssb.12162","article-title":"The normal law under linear restrictions: Simulation and estimation via minimax tilting","volume":"79","author":"Botev","year":"2017","journal-title":"J. R. Statist. Soc. B"},{"key":"2019111117080417500_B13","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1214\/09-AOAS285","article-title":"BART: Bayesian additive regression trees","volume":"4","author":"Chipman","year":"2010","journal-title":"Ann. Appl. Statist."},{"key":"2019111117080417500_B14","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1214\/lnms\/1215540964","article-title":"The practical implementation of Bayesian model selection","volume":"38","author":"Chipman","year":"2001","journal-title":"Model Selection"},{"key":"2019111117080417500_B15","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s11222-009-9168-1","article-title":"Fast simulation of truncated Gaussian distributions","volume":"21","author":"Chopin","year":"2011","journal-title":"Statist. Comp."},{"key":"2019111117080417500_B16","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1214\/16-STS581","article-title":"Leave Pima Indians alone: Binary regression as a benchmark for Bayesian computation","volume":"32","author":"Chopin","year":"2017","journal-title":"Statist. Sci."},{"key":"2019111117080417500_B17","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1111\/insr.12249","article-title":"Methods and tools for Bayesian variable selection and model averaging in normal linear regression","volume":"86","author":"Forte","year":"2018","journal-title":"Int. Statist. Rev."},{"key":"2019111117080417500_B18","doi-asserted-by":"crossref","first-page":"3509","DOI":"10.1016\/j.csda.2006.10.006","article-title":"Auxiliary mixture sampling with applications to logistic models","volume":"51","author":"Fr\u00fchwirth-Schnatter","year":"2007","journal-title":"Comp. Statist. Data Anal."},{"key":"2019111117080417500_B19","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1214\/08-AOAS191","article-title":"A weakly informative default prior distribution for logistic and other regression models","volume":"2","author":"Gelman","year":"2008","journal-title":"Ann. Appl. Statist."},{"key":"2019111117080417500_B20","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1080\/10618600.2017.1375936","article-title":"Hierarchical decompositions for the computation of high-dimensional multivariate normal probabilities","volume":"27","author":"Genton","year":"2018","journal-title":"J. Comp. Graph. Statist."},{"key":"2019111117080417500_B21","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1080\/10618600.1992.10477010","article-title":"Numerical computation of multivariate normal probabilities","volume":"1","author":"Genz","year":"1992","journal-title":"J. Comp. Graph. Statist."},{"key":"2019111117080417500_B22","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-01689-9","author":"Genz","year":"2009","journal-title":"Computation of Multivariate Normal and t Probabilities"},{"key":"2019111117080417500_B23","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.jspi.2003.09.008","article-title":"Additive properties of skew normal random vectors","volume":"126","author":"Gonz\u00e1lez-Far\u00edas","year":"2004","journal-title":"J. Statist. Plan. Infer."},{"key":"2019111117080417500_B24","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1080\/15598608.2013.790233","article-title":"On some properties of the unified skew-normal distribution","volume":"7","author":"Gupta","year":"2013","journal-title":"J. Statist. Theory Pract."},{"key":"2019111117080417500_B25","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0047-259X(03)00131-3","article-title":"A multivariate skew normal distribution","volume":"89","author":"Gupta","year":"2004","journal-title":"J. Mult. Anal."},{"key":"2019111117080417500_B26","doi-asserted-by":"crossref","first-page":"223","DOI":"10.2307\/3318737","article-title":"An adaptive Metropolis algorithm","volume":"7","author":"Haario","year":"2001","journal-title":"Bernoulli"},{"key":"2019111117080417500_B27","first-page":"382","article-title":"Bayesian model averaging: A tutorial","volume":"14","author":"Hoeting","year":"1999","journal-title":"Statist. Sci."},{"key":"2019111117080417500_B28","first-page":"1593","article-title":"The no-U-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo","volume":"15","author":"Hoffman","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"2019111117080417500_B29","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1214\/06-BA105","article-title":"Bayesian auxiliary variable models for binary and multinomial regression","volume":"1","author":"Holmes","year":"2006","journal-title":"Bayesian Anal."},{"key":"2019111117080417500_B30","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.jmva.2004.10.007","article-title":"Some results on the multivariate truncated normal distribution","volume":"94","author":"Horrace","year":"2005","journal-title":"J. Mult. Anal."},{"key":"2019111117080417500_B31","doi-asserted-by":"crossref","first-page":"1394","DOI":"10.1080\/01621459.2018.1505626","article-title":"MCMC for imbalanced categorical data","volume":"114","author":"Johndrow","year":"2019","journal-title":"J. Am. Statist. Assoc."},{"key":"2019111117080417500_B32","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1080\/01621459.1995.10476572","article-title":"Bayes factors","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Statist. Assoc."},{"key":"2019111117080417500_B33","first-page":"1679","article-title":"Assessing approximate inference for binary Gaussian process classification","volume":"6","author":"Kuss","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"2019111117080417500_B34","first-page":"72","article-title":"Exploratory analysis of cancer SAGE data","author":"Martinez","year":"2005","journal-title":"Proc. ECML\u2013PKDD Discovery Challenge Workshop (Porto, Portugal)"},{"key":"2019111117080417500_B35","doi-asserted-by":"crossref","first-page":"2740","DOI":"10.1214\/11-AOS917","article-title":"Fully Bayes factors with a generalized $g$-prior","volume":"39","author":"Maruyama","year":"2011","journal-title":"Ann. Statist."},{"key":"2019111117080417500_B36","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1214\/09-BA403","article-title":"A review of Bayesian variable selection methods: What, how and which","volume":"4","author":"O\u2019Hara","year":"2009","journal-title":"Bayesian Anal."},{"key":"2019111117080417500_B37","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1080\/01621459.2013.829001","article-title":"Bayesian inference for logistic models using P\u00f3lya\u2013Gamma latent variables","volume":"108","author":"Polson","year":"2013","journal-title":"J. Am. Statist. Assoc."},{"key":"2019111117080417500_B38","author":"Quarteroni","year":"2010","journal-title":"Numerical Mathematics"},{"key":"2019111117080417500_B39","year":"2019","journal-title":"R: A Language and Environment for Statistical Computing"},{"key":"2019111117080417500_B40","author":"Rasmussen","year":"2006","journal-title":"Gaussian Processes for Machine Learning"},{"key":"2019111117080417500_B41","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1214\/ss\/1015346320","article-title":"Optimal scaling for various Metropolis-Hastings algorithms","volume":"16","author":"Roberts","year":"2001","journal-title":"Statist. Sci."},{"key":"2019111117080417500_B42","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1111\/1467-9868.00070","article-title":"Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler","volume":"59","author":"Roberts","year":"1997","journal-title":"J. R. Statist. Soc. B"},{"key":"2019111117080417500_B43","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1214\/11-BA605","article-title":"Nonparametric Bayesian models through probit stick-breaking processes","volume":"6","author":"Rodriguez","year":"2011","journal-title":"Bayesian Anal."},{"key":"2019111117080417500_B44","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1002\/net.3230200507","article-title":"Sequential updating of conditional probabilities on directed graphical structures","volume":"20","author":"Spiegelhalter","year":"1990","journal-title":"Networks"},{"key":"2019111117080417500_B45","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1109\/ICTAI.2007.131","article-title":"Accurate classification of SAGE data based on frequent patterns of gene expression","author":"Tzanis","year":"2007","journal-title":"Proc. 19th IEEE Int. Conf. Tools with Artificial Intelligence (Patras, Greece)"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/106\/4\/765\/30646645\/asz034.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/106\/4\/765\/30646645\/asz034.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T16:51:15Z","timestamp":1721407875000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/106\/4\/765\/5554418"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,25]]},"references-count":45,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,8,25]]},"published-print":{"date-parts":[[2019,12,1]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asz034","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2019,12]]},"published":{"date-parts":[[2019,8,25]]}}}