[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-128573-en":3,"doc-seo-128573-105":31,"detail-sidebar-cat-0-en-105":96},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},128573,549768064778,"Finn","https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8",8,"Research & Report","Bayesian Quadrature for Gaussian Process Kernel Learning - Neural Ensemble Search - and High Dimensional Integrands - Thesis","Bayesian quadrature addresses marginalisation integrals that are often intractable and where integrand evaluations are costly. The approach constructs a probabilistic surrogate over the integrand and allocates evaluations using Bayesian decision-theoretic acquisition. The thesis proposes BQ schemes that exploit special parameter structure for Gaussian-process models and neural networks, and studies scalable GP approximations for higher-dimensional Euclidean spaces with non-negative integrands. It further develops GP kernel learning via maximum mean discrepancies and neural ensemble search via architecture ensembling weighted by performance. The work investigates computational complexity and scalable variational/spherical-harmonics and Bézier-curve GP approximations.","Bayesian Quadrature for Gaussian Process Kernel Learning, Neural Ensemble Search, and High Dimensional Integrands  \nSaad Hamid  \nKellogg College and Department of Engineering Science University of Oxford, Oxford, UK  \nSupervised by Prof. Michael Osborne  \nand  \nProf. Stephen Roberts  \nA thesis presented for the degree of Doctor of Philosophy  \nHilary Term 2023  \nAbstract  \nThe central challenge of performing inference in a model is the computation of marginalisation integrals over the model’s parameters. In most cases of interest, these integrals are intractable, and evaluation of the integrand is expensive. A probabilistic approach to numerical integration offers a principled framework for allocating computation in such a setting. This is achieved by using a probabilistic surrogate to model the integrand, and selecting evaluations Bayesian Decision Theoretically. We offer Bayesian Quadrature (BQ) schemes that incorporate special structure in the model parameters for two widely-used model classes: Gaussian Processes (for which we marginalise over a broad class of stationary kernels), and Neural Networks (for which we marginalise over a large space of architectures) . We further investigate the use of scalable approximations of Gaussian Processes for scaling BQ to higher dimensional (Euclidean) spaces for non-negative integrands.  \nFor GP kernel learning, our BQ framework makes use of the maximum mean discrepancies between distributions to define a kernel over kernels that captures invariances between Spectral Mixture (SM) Kernels. Kernel samples are then selected by generalising an information-theoretic acquisition function for warped BQ.  \nBy viewing ensembling as approximately marginalising over architectures, we bring the tools of BQ to bear upon Neural Ensemble Search. Additionally, the resulting ensembles consist of architectures weighted commensurately with their performance, unlike previous approaches that use equally weighted ensembles.  \nThe core challenge of scaling BQ to higher dimensions is the cubic complex-  \nity of GP regression. We explore the use of scalable approximations to GPs for BQ, particularly the recently proposed VISH model – a Variational GP for which the inter-domain inducing variables are projections of the modelled function onto the spherical harmonics – and B´ezier GP model – defined by placing a Gaussian distribution over the control points of a B´ezier curve.  \nAcknowledgements  \nI am grateful to the EPSRC (Engineering and Physical Sciences Research Council) and Kellogg College for providing the funding that allowed me to undertake this DPhil.  \nI would like to thank my primary supervisor Mike Osborne for his constant support throughout my DPhil. His encouragement fostered in me an interest in Probabilistic Numerics, and the academic freedom he provided allowed me to develop my confidence as a researcher. I thank also my co-supervisor, Steve Roberts, for his erudite guidance and his unwavering patience.  \nI owe thanks to my collaborators, Martin Jørgensen, Sebastian Schulze, Xingchen Wan, Binxin Ru, and Vincent Dutordoir, for insightful discussions and for their infectious enthusiasm. The wider BXL and MLRG also derserve my thanks for the many interesting conversations.  \nI am deeply grateful to my parents and my brothers for their love and support over these last few years.  \nFinally, I would like to thank the many friends who have been a constant source of warmth for me, and of whom Yee He and Ada Hermelink deserve special mention.  \nContents  \nList of Figures vii  \nList of Tables ix  \n1 Introduction 1  \n2 Background 5  \n2.1 Hierarchical Bayesian Modelling ..................... 5  \n2.2 Gaussian Processes ............................ 7  \n2.2.1 Scalable Gaussian Process Approximations ........... 10  \n2.2.2 Covariance Functions on non-Euclidean Spaces ......... 12  \n2.2.3 Kernel Learning .......................... 14  \n2.3 Bayesian Quadrature ........................... 16  \n2.3.1 Warped Bayesi","cbCaijKDJnDZk8Rv","https://ap.wps.com/l/cbCaijKDJnDZk8Rv","pdf",11727024,3,1,135,"English","en",105,"# Introduction\n## Background\n### Hierarchical Bayesian Modelling\n### Gaussian Processes\n## Bayesian Quadrature\n## Neural Architecture Search\n# Marginalising over Stationary Kernels for Gaussian Process Regression with Probabilistic Integration\n## Related Work\n## Our Method: MASKERADE\n## Results\n# Bayesian Quadrature for Neural Ensemble Search\n## Background","[{\"question\":\"What problem does Bayesian quadrature target in this thesis?\",\"answer\":\"It targets marginalisation integrals that are usually intractable and expensive to evaluate, by modelling the integrand probabilistically and selecting evaluation points using Bayesian decision-theoretic acquisition.\"},{\"question\":\"How does the thesis connect Bayesian quadrature with Gaussian process kernel learning?\",\"answer\":\"It uses maximum mean discrepancies between distributions to define a kernel over kernels that captures invariances between Spectral Mixture kernels, then selects kernel samples via a generalised information-theoretic acquisition for warped Bayesian quadrature.\"},{\"question\":\"How is neural ensemble search formulated using Bayesian quadrature?\",\"answer\":\"The thesis views ensembling as approximately marginalising over architectures, allowing Bayesian quadrature tools to produce ensembles whose architectures are weighted in proportion to performance rather than equally weighted.\"},{\"question\":\"Why is scaling Bayesian quadrature to high dimensions challenging, and what approximations are studied?\",\"answer\":\"The main challenge is the cubic complexity of GP regression. The thesis explores scalable GP approximations, focusing on VISH (variational GP with inducing variables projected onto spherical harmonics) and a Bézier GP model based on Gaussian control points of a Bézier curve.\"}]","Bayesian Quadrature for Gaussian Process Kernel Learning - Neural Ensemble Search - and High Dimensional Integrands - Thesis | PDF",1786001843,340,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":91,"head_meta":93,"extra_data":95,"updated_unix":29},"bayesian-quadrature-for-gaussian-process-kernel-learning-neural-ensemble-search-and-high-dimensional-integrands-thesis","",{"@graph":37,"@context":90},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,51],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":20},"https://docshare.wps.com/document/research-report/",{"item":52,"name":13,"@type":44,"position":53},"https://docshare.wps.com/document/bayesian-quadrature-for-gaussian-process-kernel-learning-neural-ensemble-search-and-high-dimensional-integrands-thesis/128573/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-25","2026-08-06",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82,86],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does Bayesian quadrature target in this thesis?","Question",{"text":76,"@type":77},"It targets marginalisation integrals that are usually intractable and expensive to evaluate, by modelling the integrand probabilistically and selecting evaluation points using Bayesian decision-theoretic acquisition.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the thesis connect Bayesian quadrature with Gaussian process kernel learning?",{"text":81,"@type":77},"It uses maximum mean discrepancies between distributions to define a kernel over kernels that captures invariances between Spectral Mixture kernels, then selects kernel samples via a generalised information-theoretic acquisition for warped Bayesian quadrature.",{"name":83,"@type":74,"acceptedAnswer":84},"How is neural ensemble search formulated using Bayesian quadrature?",{"text":85,"@type":77},"The thesis views ensembling as approximately marginalising over architectures, allowing Bayesian quadrature tools to produce ensembles whose architectures are weighted in proportion to performance rather than equally weighted.",{"name":87,"@type":74,"acceptedAnswer":88},"Why is scaling Bayesian quadrature to high dimensions challenging, and what approximations are studied?",{"text":89,"@type":77},"The main challenge is the cubic complexity of GP regression. The thesis explores scalable GP approximations, focusing on VISH (variational GP with inducing variables projected onto spherical harmonics) and a Bézier GP model based on Gaussian control points of a Bézier curve.","https://schema.org",{"og:url":52,"og:type":92,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":94,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":97},[98,102,106,110,115,120,125,128,133,136,140],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":47,"category_name":107,"show_sort_weight":108,"slug":109},"Exam",70,"exam",{"id":111,"doc_module":4,"doc_module_name":47,"category_name":112,"show_sort_weight":113,"slug":114},5,"Comic",60,"comic",{"id":116,"doc_module":4,"doc_module_name":47,"category_name":117,"show_sort_weight":118,"slug":119},6,"Technology",50,"technology",{"id":121,"doc_module":4,"doc_module_name":47,"category_name":122,"show_sort_weight":123,"slug":124},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":126,"slug":127},30,"research-report",{"id":129,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":131,"slug":132},9,"Religion & Spirituality",20,"religion-spirituality",{"id":131,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":131,"slug":135},"World Cup","world-cup",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":137,"slug":139},10,"Lifestyle","lifestyle",{"id":141,"doc_module":4,"doc_module_name":47,"category_name":142,"show_sort_weight":111,"slug":143},19,"General","general"]