{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T19:45:38Z","timestamp":1791575138071,"version":"4.3.4"},"reference-count":63,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2023,3,9]],"date-time":"2023-03-09T00:00:00Z","timestamp":1678320000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"name":"National Science Foundation","award":["DMS-2210796"],"award-info":[{"award-number":["DMS-2210796"]}]},{"name":"National Science Foundation","award":["R01ES027498 and R01ES028804"],"award-info":[{"award-number":["R01ES027498 and R01ES028804"]}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"crossref"}]},{"name":"European Research Council","award":["856506"],"award-info":[{"award-number":["856506"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,5,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>High-dimensional categorical data are routinely collected in biomedical and social sciences. It is of great importance to build interpretable parsimonious models that perform dimension reduction and uncover meaningful latent structures from such discrete data. Identifiability is a fundamental requirement for valid modeling and inference in such scenarios, yet is challenging to address when there are complex latent structures. In this article, we propose a class of identifiable multilayer (potentially deep) discrete latent structure models for discrete data, termed Bayesian Pyramids. We establish the identifiability of Bayesian Pyramids by developing novel transparent conditions on the pyramid-shaped deep latent directed graph. The proposed identifiability conditions can ensure Bayesian posterior consistency under suitable priors. As an illustration, we consider the two-latent-layer model and propose a Bayesian shrinkage estimation approach. Simulation results for this model corroborate the identifiability and estimatability of model parameters. Applications of the methodology to DNA nucleotide sequence data uncover useful discrete latent features that are highly predictive of sequence types. The proposed framework provides a recipe for interpretable unsupervised learning of discrete data and can be a useful alternative to popular machine learning methods.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkad010","type":"journal-article","created":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T21:57:55Z","timestamp":1675979875000},"page":"399-426","source":"Crossref","is-referenced-by-count":17,"title":["Bayesian Pyramids: identifiable multilayer discrete latent structure models for discrete data"],"prefix":"10.1093","volume":"85","author":[{"given":"Yuqi","family":"Gu","sequence":"first","affiliation":[{"name":"Department of Statistics, Columbia University , New York, NY , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David B","family":"Dunson","sequence":"additional","affiliation":[{"name":"Department of Statistical Science, Duke University , Durham, NC , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,3,9]]},"reference":[{"issue":"6A","key":"2023050420355870800_qkad010-B1","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":"The Annals of Statistics"},{"key":"2023050420355870800_qkad010-B2","author":"Anandkumar","year":"2013"},{"key":"2023050420355870800_qkad010-B3","author":"Anderson","year":"1956"},{"issue":"1","key":"2023050420355870800_qkad010-B4","first-page":"8753","article-title":"Learning certifiably optimal rule lists for categorical data","volume":"18","author":"Angelino","year":"2017","journal-title":"The Journal of Machine Learning Research"},{"key":"2023050420355870800_qkad010-B5","first-page":"993","article-title":"Latent Dirichlet allocation","volume":"3","author":"Blei","year":"2003","journal-title":"Journal of Machine Learning Research"},{"issue":"1","key":"2023050420355870800_qkad010-B6","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s11336-019-09693-2","article-title":"A sparse latent class model for cognitive diagnosis","volume":"85","author":"Chen","year":"2020","journal-title":"Psychometrika"},{"issue":"532","key":"2023050420355870800_qkad010-B7","doi-asserted-by":"crossref","first-page":"1756","DOI":"10.1080\/01621459.2019.1635485","article-title":"Structured latent factor analysis for large-scale data: Identifiability, estimability, and their implications","volume":"115","author":"Chen","year":"2020","journal-title":"Journal of the American Statistical Association"},{"issue":"510","key":"2023050420355870800_qkad010-B8","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1080\/01621459.2014.934827","article-title":"Statistical analysis of Q-matrix based diagnostic classification models","volume":"110","author":"Chen","year":"2015","journal-title":"Journal of the American Statistical Association"},{"key":"2023050420355870800_qkad010-B9","doi-asserted-by":"crossref","DOI":"10.1002\/9780470567333","volume-title":"Latent class and latent transition analysis: With applications in the social, behavioral, and health sciences","author":"Collins","year":"2009"},{"issue":"2","key":"2023050420355870800_qkad010-B10","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s11336-011-9207-7","article-title":"The generalized DINA model framework","volume":"76","author":"de la Torre","year":"2011","journal-title":"Psychometrika"},{"issue":"1","key":"2023050420355870800_qkad010-B11","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1023\/A:1013164120801","article-title":"On Bayesian model and variable selection using MCMC","volume":"12","author":"Dellaportas","year":"2002","journal-title":"Statistics and Computing"},{"key":"2023050420355870800_qkad010-B12","author":"Doshi-Velez","year":"2009"},{"key":"2023050420355870800_qkad010-B13","author":"Doshi-Velez","year":"2017"},{"issue":"2","key":"2023050420355870800_qkad010-B14","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1214\/10-AOS859","article-title":"Global identifiability of linear structural equation models","volume":"39","author":"Drton","year":"2011","journal-title":"The Annals of Statistics"},{"key":"2023050420355870800_qkad010-B15","author":"Dua","year":"2017"},{"issue":"487","key":"2023050420355870800_qkad010-B16","doi-asserted-by":"crossref","first-page":"1042","DOI":"10.1198\/jasa.2009.tm08439","article-title":"Nonparametric Bayes modeling of multivariate categorical data","volume":"104","author":"Dunson","year":"2009","journal-title":"Journal of the American Statistical Association"},{"issue":"suppl_1","key":"2023050420355870800_qkad010-B17","doi-asserted-by":"crossref","first-page":"5220","DOI":"10.1073\/pnas.0307760101","article-title":"Mixed-membership models of scientific publications","volume":"101","author":"Erosheva","year":"2004","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"5","key":"2023050420355870800_qkad010-B18","first-page":"109974","article-title":"The junior Eysenck personality questionnaire","volume":"169","author":"Eysenck","year":"2020","journal-title":"Personality and Individual Differences"},{"issue":"1","key":"2023050420355870800_qkad010-B19","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s11336-018-09658-x","article-title":"On the identifiability of diagnostic classification models","volume":"84","author":"Fang","year":"2019","journal-title":"Psychometrika"},{"key":"2023050420355870800_qkad010-B20","doi-asserted-by":"crossref","DOI":"10.1201\/b16018","volume-title":"Bayesian data analysis","author":"Gelman","year":"2013"},{"issue":"4","key":"2023050420355870800_qkad010-B21","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":"The Annals of Applied Statistics"},{"issue":"4","key":"2023050420355870800_qkad010-B22","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1214\/ss\/1177011136","article-title":"Inference from iterative simulation using multiple sequences","volume":"7","author":"Gelman","year":"1992","journal-title":"Statistical Science"},{"issue":"2","key":"2023050420355870800_qkad010-B23","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1093\/biomet\/61.2.215","article-title":"Exploratory latent structure analysis using both identifiable and unidentifiable models","volume":"61","author":"Goodman","year":"1974","journal-title":"Biometrika"},{"issue":"115","key":"2023050420355870800_qkad010-B24","first-page":"1","article-title":"Learning attribute patterns in high-dimensional structured latent attribute models","volume":"20","author":"Gu","year":"2019","journal-title":"Journal of Machine Learning Research"},{"issue":"4","key":"2023050420355870800_qkad010-B25","doi-asserted-by":"crossref","first-page":"2082","DOI":"10.1214\/19-AOS1878","article-title":"Partial identifiability of restricted latent class models","volume":"48","author":"Gu","year":"2020","journal-title":"Annals of Statistics"},{"issue":"2","key":"2023050420355870800_qkad010-B26","doi-asserted-by":"crossref","first-page":"542","DOI":"10.2307\/3215044","article-title":"Non-uniqueness in probabilistic numerical identification of bacteria","volume":"31","author":"Gyllenberg","year":"1994","journal-title":"Journal of Applied Probability"},{"issue":"4","key":"2023050420355870800_qkad010-B27","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1111\/j.1745-3984.1989.tb00336.x","article-title":"Using restricted latent class models to map the skill structure of achievement items","volume":"26","author":"Haertel","year":"1989","journal-title":"Journal of Educational Measurement"},{"issue":"5","key":"2023050420355870800_qkad010-B28","doi-asserted-by":"crossref","first-page":"5947","DOI":"10.4249\/scholarpedia.5947","article-title":"Deep belief networks","volume":"4","author":"Hinton","year":"2009","journal-title":"Scholarpedia"},{"issue":"7","key":"2023050420355870800_qkad010-B29","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1162\/neco.2006.18.7.1527","article-title":"A fast learning algorithm for deep belief nets","volume":"18","author":"Hinton","year":"2006","journal-title":"Neural Computation"},{"issue":"1","key":"2023050420355870800_qkad010-B30","first-page":"145","article-title":"Bayesian auxiliary variable models for binary and multinomial regression","volume":"1","author":"Holmes","year":"2006","journal-title":"Bayesian Analysis"},{"issue":"2","key":"2023050420355870800_qkad010-B31","doi-asserted-by":"crossref","first-page":"591","DOI":"10.3982\/ECTA13288","article-title":"Who should be treated? Empirical welfare maximization methods for treatment choice","volume":"86","author":"Kitagawa","year":"2018","journal-title":"Econometrica"},{"issue":"3","key":"2023050420355870800_qkad010-B32","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1137\/07070111X","article-title":"Tensor decompositions and applications","volume":"51","author":"Kolda","year":"2009","journal-title":"SIAM Review"},{"issue":"2","key":"2023050420355870800_qkad010-B33","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1214\/aoms\/1177729837","article-title":"The identification of structural characteristics","volume":"21","author":"Koopmans","year":"1950","journal-title":"The Annals of Mathematical Statistics"},{"issue":"2","key":"2023050420355870800_qkad010-B34","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/0024-3795(77)90069-6","article-title":"Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics","volume":"18","author":"Kruskal","year":"1977","journal-title":"Linear Algebra and its Applications"},{"key":"2023050420355870800_qkad010-B35","author":"Lazarsfeld","year":"1950"},{"key":"2023050420355870800_qkad010-B36","first-page":"873","article-title":"Sparse deep belief net model for visual area V2","volume":"20","author":"Lee","year":"2007","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2023050420355870800_qkad010-B37","author":"Lee","year":"2009"},{"issue":"3","key":"2023050420355870800_qkad010-B38","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1093\/biomet\/asaa008","article-title":"Bayesian cumulative shrinkage for infinite factorizations","volume":"107","author":"Legramanti","year":"2020","journal-title":"Biometrika"},{"key":"2023050420355870800_qkad010-B39","author":"Li","year":"2003"},{"key":"2023050420355870800_qkad010-B40","volume-title":"Mixture models: Inference and applications to clustering","author":"McLachlan","year":"1988"},{"issue":"4","key":"2023050420355870800_qkad010-B41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2601437","article-title":"Mdl4bmf: Minimum description length for Boolean matrix factorization","volume":"8","author":"Miettinen","year":"2014","journal-title":"ACM Transactions on Knowledge Discovery from Data (TKDD)"},{"issue":"1","key":"2023050420355870800_qkad010-B42","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1613\/jair.3879","article-title":"A survey on latent tree models and applications","volume":"47","author":"Mourad","year":"2013","journal-title":"Journal of Artificial Intelligence Research"},{"issue":"44","key":"2023050420355870800_qkad010-B43","doi-asserted-by":"crossref","first-page":"22071","DOI":"10.1073\/pnas.1900654116","article-title":"Interpretable machine learning: Definitions, methods, and applications","volume":"116","author":"Murdoch","year":"2019","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"2023050420355870800_qkad010-B44","volume-title":"Machine learning: A probabilistic perspective","author":"Murphy","year":"2012"},{"issue":"5","key":"2023050420355870800_qkad010-B45","doi-asserted-by":"crossref","first-page":"280","DOI":"10.4236\/jbise.2016.95021","article-title":"DNA sequence classification by convolutional neural network","volume":"9","author":"Nguyen","year":"2016","journal-title":"Journal of Biomedical Science and Engineering"},{"key":"2023050420355870800_qkad010-B46","doi-asserted-by":"crossref","DOI":"10.1017\/9781108591720","volume-title":"Joint species distribution modelling: With applications in R","author":"Ovaskainen","year":"2020"},{"issue":"5","key":"2023050420355870800_qkad010-B47","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1111\/2041-210X.12501","article-title":"Using latent variable models to identify large networks of species-to-species associations at different spatial scales","volume":"7","author":"Ovaskainen","year":"2016","journal-title":"Methods in Ecology and Evolution"},{"issue":"1","key":"2023050420355870800_qkad010-B48","first-page":"1","article-title":"Label.switching: An R package for dealing with the label switching problem in MCMC outputs","volume":"69","author":"Papastamoulis","year":"2016","journal-title":"Journal of Statistical Software"},{"key":"2023050420355870800_qkad010-B49","volume-title":"Probabilistic reasoning in intelligent systems: Networks of plausible inference","author":"Pearl","year":"2014"},{"issue":"4","key":"2023050420355870800_qkad010-B50","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1016\/j.cell.2005.06.026","article-title":"Genome-wide map of nucleosome acetylation and methylation in yeast","volume":"122","author":"Pokholok","year":"2005","journal-title":"Cell"},{"issue":"504","key":"2023050420355870800_qkad010-B51","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":"Journal of the American Statistical Association"},{"key":"2023050420355870800_qkad010-B52","author":"Poon","year":"2011"},{"issue":"5","key":"2023050420355870800_qkad010-B53","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":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"issue":"5","key":"2023050420355870800_qkad010-B54","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1038\/s42256-019-0048-x","article-title":"Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead","volume":"1","author":"Rudin","year":"2019","journal-title":"Nature Machine Intelligence"},{"issue":"4","key":"2023050420355870800_qkad010-B55","first-page":"219","article-title":"Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art","volume":"6","author":"Rupp","year":"2008","journal-title":"Measurement"},{"issue":"2","key":"2023050420355870800_qkad010-B56","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1093\/ectj\/utaa027","article-title":"Debiased machine learning of conditional average treatment effects and other causal functions","volume":"24","author":"Semenova","year":"2021","journal-title":"The Econometrics Journal"},{"issue":"S1","key":"2023050420355870800_qkad010-B57","doi-asserted-by":"crossref","first-page":"S64","DOI":"10.1111\/insr.12285","article-title":"Analysis of categorical data for complex surveys","volume":"87","author":"Skinner","year":"2019","journal-title":"International Statistical Review"},{"issue":"1","key":"2023050420355870800_qkad010-B58","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1214\/aoms\/1177705155","article-title":"Identifiability of mixtures","volume":"32","author":"Teicher","year":"1961","journal-title":"The Annals of Mathematical Statistics"},{"key":"2023050420355870800_qkad010-B59","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-05584-4","volume-title":"Handbook of diagnostic classification models","author":"von Davier","year":"2019"},{"issue":"2","key":"2023050420355870800_qkad010-B60","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1214\/16-AOS1464","article-title":"Identifiability of restricted latent class models with binary responses","volume":"45","author":"Xu","year":"2017","journal-title":"The Annals of Statistics"},{"key":"2023050420355870800_qkad010-B61","author":"Zhao","year":"2015"},{"issue":"512","key":"2023050420355870800_qkad010-B62","doi-asserted-by":"crossref","first-page":"1562","DOI":"10.1080\/01621459.2014.983233","article-title":"Bayesian factorizations of big sparse tensors","volume":"110","author":"Zhou","year":"2015","journal-title":"Journal of the American Statistical Association"},{"key":"2023050420355870800_qkad010-B63","author":"Zwiernik","year":"2018"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article-pdf\/doi\/10.1093\/jrsssb\/qkad010\/49481831\/qkad010.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/85\/2\/399\/50204620\/qkad010.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/85\/2\/399\/50204620\/qkad010.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,4]],"date-time":"2023-05-04T16:38:34Z","timestamp":1683218314000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/85\/2\/399\/7074362"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,9]]},"references-count":63,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,3,9]]},"published-print":{"date-parts":[[2023,5,4]]}},"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkad010","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2023,4,1]]},"published":{"date-parts":[[2023,3,9]]}}}