[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125459-en":3,"doc-seo-125459-105":30,"detail-sidebar-cat-0-en-105":91},{"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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},125459,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Multivariate Latent Recalibration for Conditional Normalizing Flows - Research","Reliable estimation of the full conditional distribution of a multivariate response given covariates is critical for trustworthy decision-making, yet misspecified or miscalibrated models produce poor joint approximations and unreliable predictions. Existing recalibration approaches mainly target univariate cases and do not yield an explicit multivariate probability distribution. The method introduces latent calibration and a post-hoc latent recalibration procedure for invertible generative models such as normalizing flows, providing latent calibration guarantees, an explicit density, and efficient density evaluation. Experiments show consistent improvements in latent calibration error and negative log-likelihood across tabular and image data.","Multivariate Latent Recalibration for Conditional Normalizing Flows  \nVictor Dheur  \nDepartment of Computer Science University of Mons Mons, Belgium [victor.dheur@umons.ac.be](victor.dheur@umons.ac.be)  \nSouhaib Ben Taieb∗  \nDepartment of Statistics and Data Science Mohamed bin Zayed University of Artificial Intelligence Abu Dhabi, United Arab Emirates [souhaib.bentaieb@mbzuai.ac.ae](souhaib.bentaieb@mbzuai.ac.ae)  \nAbstract  \nA reliable estimate of the full conditional distribution of a multivariate response given a set of covariates is essential in many decision-making applications. However, misspecified or miscalibrated models can lead to poor approximations of the joint distribution, resulting in unreliable predictions and suboptimal decisions.  \nStandard recalibration methods are largely restricted to univariate settings, and while conformal prediction techniques yield multivariate regions with coverage guarantees, they do not provide an explicit form of the underlying probability distribution. We address this gap by first introducing a novel notion of latent calibration, which assesses probabilistic calibration in the latent space of conditional invertible generative models such as normalizing flows and flow matching. Second, we propose latent recalibration (LR), a post-hoc model recalibration method that learns a transformation of the latent space with finite-sample bounds on latent calibration.  \nUnlike existing recalibration methods, LR produces a recalibrated distribution with an explicit multivariate density function while remaining computationally efficient. Extensive experiments on both tabular and image datasets show that LR consistently improves latent calibration error and the negative log-likelihood of therecalibrated models.  \n1 Introduction  \nGenerating reliable uncertainty estimates is essential for trustworthy decision-making across a wide range of applications (Gawlikowski et al., 2023) . Multi-output regression problems, in particular, arise frequently in domains such as weather forecasting (Setiawan et al., 2024), energy consumption prediction (Makaremi, 2025), and healthcare resource utilization (Cui et al., 2018) . While flexible models like neural networks can achieve high predictive accuracy, their uncertainty estimates are often poorly calibrated, meaning predicted probabilities or confidence regions do not align with empirical frequencies (Guo et al., 2017; Dheur and Ben Taieb, 2023) . Furthermore, most recalibration methods are designed for the single-output setting (Gneiting et al., 2007; Song et al., 2019; Sahoo et al., 2021; Kuleshov and Deshpande, 2021; Dewolf et al., 2022; Fakoor et al., 2023; Marx et al., 2023; Chung et al., 2023; Gneiting and Resin, 2023) .  \nNoting the general lack of methods for assessing and recalibrating multi-output models, Chung et al.(2024) leveraged highest-density regions (HDRs) (Hyndman, 1996) to introduce the notion of HDR calibration and propose the sampling-based HDR recalibration (HDR-R) method. Recently, multioutput conformal prediction (CP) methods have also been developed to construct joint prediction sets (Wang et al., 2023; Feldman et al., 2023; Fang et al., 2025; Dheur et al., 2025) . However, both HDR-R and CP approaches fail to provide an explicit form for the underlying recalibrated probability  \n∗Also affiliated with the Department of Computer Science, University of Mons.  \n39th Conference on Neural Information Processing Systems (NeurIPS 2025) .  \ndistribution, with HDR-R further involving computationally intensive sampling and binning at test time.  \nTo overcome these limitations, we introduce the latent recalibration (LR) method, based on a new notion of latent calibration, which recalibrates invertible generative models (e.g., normalizing flows (NFs) or flow matching (FM)) by operating within their latent space. The core idea is to learn a transformation of the latent space such that the resulting model achieves latent calibration. Compared to s","cbCaiazcpotDszoe","https://ap.wps.com/l/cbCaiazcpotDszoe","pdf",8458086,1,36,"English","en",105,"# Introduction\n# Background\n## Multi-output distributional regression setup\n## Datasets and conditional distribution notation\n# Method and contributions\n## Latent calibration concept\n## Latent recalibration (LR) procedure\n# Experimental evaluation\n## Tabular and image datasets\n# Conclusion","[{\"question\":\"Why is recalibration important for multivariate conditional prediction?\",\"answer\":\"Miscalibrated models can misrepresent the joint distribution of multivariate outputs, causing unreliable predictions and suboptimal decisions.\"},{\"question\":\"What gap does latent recalibration address compared with existing methods?\",\"answer\":\"Standard recalibration is mostly univariate, while conformal approaches provide coverage regions but not an explicit form of the underlying multivariate probability distribution.\"},{\"question\":\"How does LR remain explicit and computationally efficient?\",\"answer\":\"LR operates in the latent space of invertible generative models, learns a latent transformation for latent calibration, and produces a recalibrated distribution with an explicit multivariate density while enabling efficient density evaluation and sampling.\"}]","Multivariate Latent Recalibration for Conditional Normalizing Flows - Research | PDF",1785899125,91,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"multivariate-latent-recalibration-for-conditional-normalizing-flows-research","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/multivariate-latent-recalibration-for-conditional-normalizing-flows-research/125459/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is recalibration important for multivariate conditional prediction?","Question",{"text":75,"@type":76},"Miscalibrated models can misrepresent the joint distribution of multivariate outputs, causing unreliable predictions and suboptimal decisions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What gap does latent recalibration address compared with existing methods?",{"text":80,"@type":76},"Standard recalibration is mostly univariate, while conformal approaches provide coverage regions but not an explicit form of the underlying multivariate probability distribution.",{"name":82,"@type":73,"acceptedAnswer":83},"How does LR remain explicit and computationally efficient?",{"text":84,"@type":76},"LR operates in the latent space of invertible generative models, learns a latent transformation for latent calibration, and produces a recalibrated distribution with an explicit multivariate density while enabling efficient density evaluation and sampling.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]