[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82444-en":3,"doc-seo-82444-105":29,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82444,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Deep Gaussian Processes on Directed Acyclic Graphs","Compositions of partially observed functions across directed acyclic graphs (DAGs) arise in causal modelling, engineering multi-fidelity systems, and gene-regulatory networks. Noisy, heterogeneously sampled measurements create coupled inverse inference, difficult uncertainty propagation, and explaining-away effects. The work introduces deep Gaussian process priors tailored to DAG structure, studies prior-collapse behavior, derives almost-sure asymptotic information-preservation bounds, and proposes a structured variational approximation. Empirical validation models latent-collider, protein signalling, and multi-fidelity heavy-ion emulation with state-of-the-art performance and interpretability.","arXiv :2607 .09645v1 [ stat .ML] 10 Jul 2026  \nDeep Gaussian Processes on Directed Acyclic Graphs  \nFederico L. Perlino 1 Oliver Hamelijnck2 Adam M. Johansen 1 Theodoros Damoulas3,1,2  \n1 Department of Statistics, University of Warwick  \n2 AI Team, Unilink Software Ltd  \n3 Department of Computer Science, University of Warwick  \nAbstract  \nMany real-world processes can be represented as compositions of functions along a directed acyclic graph (DAG) . In causal modelling, these correspond to the underlying mechanisms; in engineering, to multiple fidelity levels; and in gene-regulatory networks, to transcription factors. These functions are partially observed across the DAG, with noisy and heterogeneously sampled measurements, posing significant challenges for reconstruction, uncertainty propagation, and inference. To tackle these challenges, we place priors over functions and naturally arrive at Deep Gaussian Processes over DAGs. We theoretically study their prior-collapse behaviour, and the effect of graph topology and intermediate observations on the preservation of information. We obtain almost-sure lower bounds on the asymptotic frequency of depthsat which the distinction between inputs is preserved, identify broad kernel classes for which these hold, and prove an observation by Dunlop et al. (2018) on the role of input connections. We offer a structured variational approximation that retains graph dependencies, preserves compositional uncertainty, and captures the explaining-away behaviour of colliders. Finally, we empirically validate our theoretical results and our methodology, and model a latent-collider DAG, a protein signalling network, and a multi-fidelity heavy-ion collision emulation task, attaining state-of-the-art performance while recovering low-fidelity contributions and yielding interpretability over the simulator hierarchy.  \n1 Introduction  \nVarious phenomena across the sciences, and beyond, can be represented as compositions of interdependent latent functions along a Directed Acyclic Graph (DAG) . In probabilistic modelling, the DAG encodes conditional dependencies between quantities of interest, although the interpretation of these dependencies varies by setting. In causal models, it represents mechanistic relations (Spirtes et al. , 2001 ; Pearl, 2009); in multi-fidelity modelling across physics and engineering (Fernández-Godino, 2023), it encodes dependencies between information sources of different fidelity (Kennedy and O’Hagan, 2000 ; Perdikaris et al. , 2017 ; Ji et al. , 2024); and in systems biology it is used to describe regulatory links between transcription factors in generegulatory networks (Friedman, 2004) . The graph itself may be elicited from expert knowledge (Del Sagrado and Moral, 2003 ; O’Hagan et al. , 2006), derived from mechanistic constraints and the natural directionality of the phenomenon (Boudali and Bechta Dugan, 2005), or inferred from data through structure discovery (Heckerman et al. , 1995) . More broadly, DAGs are often formulated by scientists as explicit representations of their hypotheses (Greenland et al. , 1999) .  \nIn such DAG settings, we rarely have complete, noise-free observations at every node. Data may be available only at a subset of nodes, at varying sample sizes and resolutions, and are often affected by missingness, measurement error, or model discrepancy (Little and Rubin, 2019 ; Carroll et al. , 2006 ; Kennedy and O’Hagan, 2001) . Inference then becomes a coupled inverse problem, in which DAG-dependent latent functions at different nodes must be jointly recovered from indirect, heterogeneous evidence. Furthermore, when an observed downstream quantity can be explained by several upstream functions, evidence for one explanation changes the posterior plausibility of the others, which is the classical explaining-away effect (Pearl, 1988 ; Lauritzen, 1996) . At the same time, latent quantities that are never directly observed are typically weakly or non-identifiab","cbCaiiOMCeKq4zm4","https://ap.wps.com/l/cbCaiiOMCeKq4zm4","pdf",1778947,1,75,"English","en",105,"# Abstract\n# Introduction\n## DAG-based compositional probabilistic modeling\n## Challenges: partial observations and explaining-away\n## Deep Gaussian processes on DAGs (DAG-DGPs)\n## Relation to multi-fidelity and information-fusion models","[{\"question\":\"Which approximation and validation approaches are proposed?\",\"answer\":\"A structured variational approximation is introduced to retain graph dependencies, preserve compositional uncertainty, and model explaining-away behavior of colliders, then validated on latent-collider, protein signalling, and multi-fidelity heavy-ion emulation tasks.\"}]",1784180409,189,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":27},"deep-gaussian-processes-on-directed-acyclic-graphs","",{"@graph":35,"@context":77},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/deep-gaussian-processes-on-directed-acyclic-graphs/82444/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Which approximation and validation approaches are proposed?","Question",{"text":75,"@type":76},"A structured variational approximation is introduced to retain graph dependencies, preserve compositional uncertainty, and model explaining-away behavior of colliders, then validated on latent-collider, protein signalling, and multi-fidelity heavy-ion emulation tasks.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]