{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:19:37Z","timestamp":1760059177889,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T00:00:00Z","timestamp":1748304000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangzhou Huashang College Project:","award":["2023HSDS25","1242005"],"award-info":[{"award-number":["2023HSDS25","1242005"]}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["2023HSDS25","1242005"],"award-info":[{"award-number":["2023HSDS25","1242005"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>For non-ignorable missing response variables, the mechanism of whether the response variable is missing can be modeled through logistic regression. In Bayesian computation, the lack of a conjugate prior for the logistic function poses a significant challenge. Introducing a new P\u00f3lya-Gamma variable and employing lower-bound approximation are two common methods for parameter inference in conjugate Bayesian logistic regression. It can be observed that these two methods yield essentially the same variational posterior in the calculation of the variational Bayesian posterior. This paper applies a popular Bayesian spike-and-slab LASSO prior for variable selection in quantile regression with non-ignorable missing response variables, which demonstrates good performance in both simulations and practical applications.<\/jats:p>","DOI":"10.3390\/axioms14060408","type":"journal-article","created":{"date-parts":[[2025,5,27]],"date-time":"2025-05-27T11:12:57Z","timestamp":1748344377000},"page":"408","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Variational Bayesian Quantile Regression with Non-Ignorable Missing Response Data"],"prefix":"10.3390","volume":"14","author":[{"given":"Juanjuan","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Digital Economy and Trade, Guangzhou Huashang College, Guangzhou 511300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-2812-2616","authenticated-orcid":false,"given":"Weixian","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0515-4477","authenticated-orcid":false,"given":"Maozai","family":"Tian","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Xinjiang University of Finance and Economics, Urumqi 830012, China"},{"name":"Center for Applied Statistics, School of Statistics, Renmin University of China, Beijing 100872, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1093\/biomet\/63.3.581","article-title":"Inference and missing data","volume":"63","author":"Rubin","year":"1976","journal-title":"Biometrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1109\/TPAMI.1984.4767596","article-title":"Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images","volume":"6","author":"Geman","year":"1984","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hastings, W.K. (1970). Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Oxford University Press.","DOI":"10.2307\/2334940"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equation of State Calculations by Fast Computing Machines","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1198\/016214504000001844","article-title":"Missing-Data Methods for Generalized Linear Models: A Comparative Review","volume":"100","author":"Ibrahim","year":"2005","journal-title":"J. Am. Stat. 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Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1080\/01621459.2016.1260469","article-title":"The spike-and-slab lasso","volume":"113","author":"George","year":"2018","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_10","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. Stat. Assoc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1023\/A:1008932416310","article-title":"Bayesian parameter estimation via variational methods","volume":"10","author":"Jaakkola","year":"2000","journal-title":"Stat. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Li, X., Tuerde, M., and Hu, X. (2023). Variational Bayesian Inference for Quantile Regression Models with Nonignorable Missing Data. 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