{"id":"https://openalex.org/W7213655982","doi":"https://doi.org/10.1002/wics.70076","title":"An Introduction to Stochastic Deep Learning","display_name":"An Introduction to Stochastic Deep Learning","publication_year":2026,"publication_date":"2026-09-01","ids":{"openalex":"https://openalex.org/W7213655982","doi":"https://doi.org/10.1002/wics.70076","pmid":"https://pubmed.ncbi.nlm.nih.gov/42959612","pmcid":"https://www.ncbi.nlm.nih.gov/pmc/articles/13717249"},"language":"en","primary_location":{"id":"doi:10.1002/wics.70076","doi":"https://doi.org/10.1002/wics.70076","is_oa":true,"landing_page_url":"https://doi.org/10.1002/wics.70076","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/wics.70076","source":{"id":"https://openalex.org/S206848516","display_name":"Wiley Interdisciplinary Reviews Computational Statistics","issn_l":"1939-0068","issn":["1939-0068","1939-5108"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-d","jufo-1","norway-1","poland-100","russia-white-list-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"WIREs Computational Statistics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/wics.70076","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085287370","display_name":"Faming Liang","orcid":"https://orcid.org/0000-0002-1177-5501","observed_orcids":["https://orcid.org/0000-0002-1177-5501"]},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Faming Liang","raw_affiliation_strings":["Department of Statistics Purdue University  West Lafayette Indiana USA"],"raw_orcid":"https://orcid.org/0000-0002-1177-5501","affiliations":[{"raw_affiliation_string":"Department of Statistics Purdue University  West Lafayette Indiana USA","institution_ids":["https://openalex.org/I219193219"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5085287370"],"corresponding_institution_ids":["https://openalex.org/I219193219"],"apc_list":{"value":3460,"currency":"USD","value_usd":3460},"apc_paid":{"value":3460,"currency":"USD","value_usd":3460},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.61616214,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"18","issue":"3","first_page":"e70076","last_page":"e70076"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.6538000106811523,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.6538000106811523,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.25049999356269836,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.01810000091791153,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/stochastic-neural-networks","display_name":"stochastic neural networks","score":0.8050000071525574},{"id":"https://openalex.org/keywords/probabilistic-latent-variable-models","display_name":"probabilistic latent variable models","score":0.8700000047683716},{"id":"https://openalex.org/keywords/deep-neural-network","display_name":"deep neural network","score":0.7490000128746033},{"id":"https://openalex.org/keywords/statistical-inference","display_name":"statistical inference","score":0.7639999985694885},{"id":"https://openalex.org/keywords/causal-inference","display_name":"causal inference","score":0.8980000019073486},{"id":"https://openalex.org/keywords/unmeasured-confounding","display_name":"unmeasured confounding","score":0.8410000205039978},{"id":"https://openalex.org/keywords/granger-causality","display_name":"Granger causality","score":0.9829999804496765},{"id":"https://openalex.org/keywords/nonlinear-time-series","display_name":"nonlinear time series","score":0.9940000176429749},{"id":"https://openalex.org/keywords/uncertainty-quantification","display_name":"uncertainty quantification","score":0.9049999713897705},{"id":"https://openalex.org/keywords/overparameterized-neural-networks","display_name":"overparameterized neural networks","score":0.7210000157356262},{"id":"https://openalex.org/keywords/representation-learning","display_name":"representation learning","score":0.6420000195503235},{"id":"https://openalex.org/keywords/training-algorithms","display_name":"training algorithms","score":0.5329999923706055}],"concepts":[{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.7156999707221985},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.685699999332428},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.6484000086784363},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.6187999844551086},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5788999795913696},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5769000053405762},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.5697000026702881},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5116000175476074},{"id":"https://openalex.org/C134261354","wikidata":"https://www.wikidata.org/wiki/Q938438","display_name":"Statistical inference","level":2,"score":0.4999000132083893},{"id":"https://openalex.org/C49937458","wikidata":"https://www.wikidata.org/wiki/Q2599292","display_name":"Probabilistic logic","level":2,"score":0.4828999936580658},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.4699000120162964},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.4399000108242035},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.39250001311302185},{"id":"https://openalex.org/C51167844","wikidata":"https://www.wikidata.org/wiki/Q4422623","display_name":"Latent variable","level":2,"score":0.3772999942302704},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3418999910354614},{"id":"https://openalex.org/C55479107","wikidata":"https://www.wikidata.org/wiki/Q97663916","display_name":"Stochastic approximation","level":3,"score":0.33399999141693115},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.32749998569488525},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.3260999917984009},{"id":"https://openalex.org/C91873725","wikidata":"https://www.wikidata.org/wiki/Q3445816","display_name":"Function approximation","level":3,"score":0.3215000033378601},{"id":"https://openalex.org/C22367795","wikidata":"https://www.wikidata.org/wiki/Q7625208","display_name":"Structured prediction","level":2,"score":0.28690001368522644},{"id":"https://openalex.org/C158600405","wikidata":"https://www.wikidata.org/wiki/Q5054566","display_name":"Causal inference","level":2,"score":0.2669999897480011},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2662999927997589},{"id":"https://openalex.org/C137002209","wikidata":"https://www.wikidata.org/wiki/Q898521","display_name":"Hidden variable theory","level":3,"score":0.2605000138282776}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1002/wics.70076","doi":"https://doi.org/10.1002/wics.70076","is_oa":true,"landing_page_url":"https://doi.org/10.1002/wics.70076","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/wics.70076","source":{"id":"https://openalex.org/S206848516","display_name":"Wiley Interdisciplinary Reviews Computational Statistics","issn_l":"1939-0068","issn":["1939-0068","1939-5108"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-d","jufo-1","norway-1","poland-100","russia-white-list-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"WIREs Computational Statistics","raw_type":"journal-article"},{"id":"pmh:oai:pubmedcentral.nih.gov:13717249","doi":null,"is_oa":true,"landing_page_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC13717249/","pdf_url":"https://pmc.ncbi.nlm.nih.gov/articles/PMC13717249/pdf/nihms-2218781.pdf","source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Wiley Interdiscip Rev Comput Stat","raw_type":"Text"},{"id":"pmid:42959612","doi":null,"is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/42959612","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Wiley interdisciplinary reviews. Computational statistics","raw_type":"Journal Article"}],"best_oa_location":{"id":"doi:10.1002/wics.70076","doi":"https://doi.org/10.1002/wics.70076","is_oa":true,"landing_page_url":"https://doi.org/10.1002/wics.70076","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/wics.70076","source":{"id":"https://openalex.org/S206848516","display_name":"Wiley Interdisciplinary Reviews Computational Statistics","issn_l":"1939-0068","issn":["1939-0068","1939-5108"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","cwts-core","dongbi-d","jufo-1","norway-1","poland-100","russia-white-list-1"],"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"WIREs Computational Statistics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G6014979109","display_name":"A New Stochastic Neural Network: Statistical Perspectives and Applications","funder_award_id":"2210819","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G9000903830","display_name":"Stochastic Deep Learning for Electronic Health Records: Localizing Learning with Massive and Fragmented Data","funder_award_id":"R01GM152717","funder_id":"https://openalex.org/F4320337354","funder_display_name":"National Institute of General Medical Sciences"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320332161","display_name":"National Institutes of Health","ror":"https://ror.org/01cwqze88"},{"id":"https://openalex.org/F4320337354","display_name":"National Institute of General Medical Sciences","ror":"https://ror.org/04q48ey07"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213655982.pdf","grobid_xml":"https://content.openalex.org/works/W7213655982.grobid-xml"},"referenced_works_count":48,"referenced_works":["https://openalex.org/W1567512734","https://openalex.org/W1965026065","https://openalex.org/W1978930294","https://openalex.org/W2019459021","https://openalex.org/W2030219534","https://openalex.org/W2036163871","https://openalex.org/W2039811614","https://openalex.org/W2041701373","https://openalex.org/W2053681597","https://openalex.org/W2068752849","https://openalex.org/W2072460512","https://openalex.org/W2074682976","https://openalex.org/W2086014844","https://openalex.org/W2087717467","https://openalex.org/W2100495367","https://openalex.org/W2116064496","https://openalex.org/W2123993001","https://openalex.org/W2130606707","https://openalex.org/W2132917208","https://openalex.org/W2135046866","https://openalex.org/W2136922672","https://openalex.org/W2143891888","https://openalex.org/W2144405862","https://openalex.org/W2154080901","https://openalex.org/W2163490846","https://openalex.org/W2171050905","https://openalex.org/W2178225550","https://openalex.org/W2526460638","https://openalex.org/W2550848904","https://openalex.org/W2782893163","https://openalex.org/W2891467507","https://openalex.org/W2973682480","https://openalex.org/W2997942693","https://openalex.org/W3124120302","https://openalex.org/W3131977067","https://openalex.org/W3134482315","https://openalex.org/W3201741216","https://openalex.org/W3204920318","https://openalex.org/W3210956727","https://openalex.org/W4220678908","https://openalex.org/W4242484299","https://openalex.org/W4288714699","https://openalex.org/W4392544739","https://openalex.org/W4417056646","https://openalex.org/W7133234816","https://openalex.org/W7133244101","https://openalex.org/W7161285531","https://openalex.org/W7167062618"],"related_works":["https://openalex.org/W4220678908","https://openalex.org/W7213362362","https://openalex.org/W6889142880","https://openalex.org/W4412814028","https://openalex.org/W6903268216","https://openalex.org/W4376653673","https://openalex.org/W4323657198","https://openalex.org/W3109737792","https://openalex.org/W25770843","https://openalex.org/W4417098554"],"abstract_inverted_index":{"Deep":[0],"neural":[1,24],"networks":[2],"(DNNs)":[3],"have":[4],"achieved":[5],"remarkable":[6],"success":[7],"in":[8,38,166,174,182],"prediction,":[9],"but":[10],"their":[11],"deterministic":[12],"formulation":[13,57],"makes":[14],"many":[15],"statistical":[16,98,211],"inference":[17,99,107],"tasks":[18],"difficult.":[19],"StoNet,":[20],"short":[21],"for":[22,84,91,115,126,210],"stochastic":[23],"network,":[25],"addresses":[26],"this":[27],"limitation":[28],"by":[29],"reformulating":[30],"a":[31,34,65,75,207],"DNN":[32],"as":[33,50,74,80,146],"probabilistic":[35],"latent-variable":[36],"model,":[37],"which":[39,142],"the":[40,56,152,159,177,186],"outputs":[41],"of":[42,58,151],"selected":[43],"hidden":[44],"layers":[45],"or":[46,148],"units":[47],"are":[48],"treated":[49],"latent":[51],"variables.":[52],"The":[53,129],"article":[54,130],"describes":[55],"StoNet":[59,70,92,137,153,198],"and":[60,79,93,110,122,138,168],"its":[61,95],"asymptotic":[62],"equivalence":[63],"to":[64,97,163,170,206],"conventional":[66,172],"DNN,":[67],"showing":[68],"how":[69],"can":[71,143],"serve":[72],"both":[73],"deep":[76,200],"learning":[77,114,201],"model":[78],"an":[81],"analytical":[82],"device":[83],"studying":[85],"DNNs.":[86,128],"It":[87],"reviews":[88],"training":[89,167,187],"algorithms":[90],"discusses":[94,132],"applications":[96],"problems,":[100],"including":[101],"nonlinear":[102,116,119],"sufficient":[103],"dimension":[104],"reduction,":[105],"causal":[106],"with":[108],"observed":[109],"unobserved":[111],"confounders,":[112],"Granger-causality":[113],"time":[117],"series,":[118],"variable":[120],"selection,":[121],"prediction":[123],"uncertainty":[124],"quantification":[125],"large-scale":[127],"also":[131],"two":[133],"related":[134],"architectures:":[135],"kernel-expanded":[136],"sublinearly":[139],"structured":[140],"DNNs,":[141],"be":[144],"viewed":[145],"structural":[147],"theoretical":[149],"extensions":[150],"framework.":[154],"Under":[155],"suitable":[156],"architectural":[157],"conditions,":[158],"former":[160],"is":[161,190],"immune":[162],"local":[164],"traps":[165],"tends":[169],"outperform":[171],"DNNs":[173],"prediction;":[175],"whereas":[176],"latter":[178],"achieves":[179],"feature-learning":[180],"consistency":[181],"over-parameterized":[183],"regimes":[184],"when":[185],"sample":[188],"size":[189],"sufficiently":[191],"large.":[192],"Together,":[193],"these":[194],"developments":[195],"show":[196],"that":[197],"broadens":[199],"from":[202],"prediction-oriented":[203],"function":[204],"approximation":[205],"richer":[208],"framework":[209],"inference.":[212]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-20T00:00:00"}
