[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117741-en":3,"doc-seo-117741-105":30,"detail-sidebar-cat-0-en-105":92},{"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":27,"seo_description":14,"update_tm":28,"read_time":29},117741,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A Review of Probabilistic Predictive Uncertainty Estimation with Machine Learning - Review","Predictions and forecasts from machine learning models can be expressed as probability distributions to increase the information conveyed to end users. While probabilistic prediction and forecasting applications are becoming common across academia and industry, the related concepts and methods lack a unified, structured overview. This review summarizes predictive uncertainty estimation with machine learning, emphasizing assessment via consistent scoring functions and proper scoring rules, and traces developments from Bayesian and quantile-regression time-series approaches to modern flexible models.","A review of probabilistic forecasting and prediction with machine learning  \nHristos Tyralis1,2,*, Georgia Papacharalampous3  \n1Construction Agency, Hellenic Air Force, Mesogion Avenue 227–231, 15 561 Cholargos, Greece ([montchrister@gmail.com](montchrister@gmail.com), [https://orcid.org/0000-0002-8932-4997](https://orcid.org/0000-0002-8932-4997))  \n2Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece (hristos@itia.ntua.gr)  \n3Department of Water Resources and Environmental Modeling, Faculty of Environmental Sciences, Czech University of Life Sciences, Kamýcá 129, Praha-Suchdol 16500, Prague, Czech Republic ([papacharalampous.georgia@gmail.com](papacharalampous.georgia@gmail.com), [https://orcid.org/0000-0001-](https://orcid.org/0000-0001-)[ ](https://orcid.org/0000-0001-)[5446-954X](5446-954X))  \n*Corresponding author  \nAbstract: Predictions and forecasts of machine learning models should take the form of probability distributions, aiming to increase the quantity of information communicated to end users. Although applications of probabilistic prediction and forecasting with machine learning models in academia and industry are becoming more frequent, related concepts and methods have not been formalized and structured under a holistic view of the entire field. Here, we review the topic of predictive uncertainty estimation with machine learning algorithms, as well as the related metrics (consistent scoring functions and proper scoring rules) for assessing probabilistic predictions. The review covers a time period spanning from the introduction of early statistical (linear regression and time series models, based on Bayesian statistics or quantile regression) to recent machine learning algorithms (including generalized additive models for location, scale and shape, random forests, boosting and deep learning algorithms) that are more flexible by nature. The review of the progress in the field, expedites our understanding on how to develop new algorithms tailored to users’ needs, since the latest advancements are based on some fundamental concepts applied to more complex algorithms. We conclude by classifying the material and discussing challenges that are becoming a hot topic of research.  \nKeywords: Boosting; deep learning; distributional regression; ensemble learning; machine learning; predictive uncertainty; quantile regression; random forests  \n1. Introduction  \nThe vast majority of supervised learning applications is related to point predictions that are intended to be close to continuous process realizations. That is possible, for instance, by optimizing machine learning regression algorithms with respect to a squared error (or similar) loss function. Although point predictions are useful, the information content can be increased when predictions take the form of probability distributions (Gneiting and Raftery 2007) .  \nNotwithstanding that statistical modelling is mostly associated with inference of parameters, it has been indicated that the ultimate goal should be the prediction of future events (Billheimer 2019) . The earliest formal strategy for estimating the probability distribution of predictions consists of fitting a Bayesian statistical model to given data (Roberts 1965, Barbieri 2015). The Bayesian problem is formulated by assigning a prior distribution to the parameters of the statistical model, updating the distribution of the parameters conditional on the data and estimating the predictive uncertainty by integrating the parameters posterior distribution. The probability distribution of the predictions is called predictive distribution, while the procedure can be called predictive uncertainty estimation (or quantification), since probability distributions characterize the uncertainty of the predictions. Although earlier considerations on predictive uncertainty estimation were Bayesian stat","cbCaiuhiyCSht6OK","https://ap.wps.com/l/cbCaiuhiyCSht6OK","pdf",1827155,1,83,"English","en",105,"# Introduction\n## Point predictions vs probabilistic predictions\n## Bayesian approaches and predictive distributions\n## Loss functions, scoring rules, and quantile loss\n## Machine learning paradigm and cross-validation","[{\"question\":\"Why should machine learning predictions be represented as probability distributions?\",\"answer\":\"Probability distributions communicate uncertainty and increase the amount of information provided to end users compared with single point estimates.\"},{\"question\":\"How does Bayesian modeling support predictive uncertainty estimation?\",\"answer\":\"Bayesian methods assign priors to model parameters, update them using data, and compute predictive uncertainty by integrating over the posterior distribution of parameters.\"},{\"question\":\"What role do scoring functions and proper scoring rules play in evaluating probabilistic predictions?\",\"answer\":\"They provide principled metrics for assessing probabilistic forecasts, supporting consistent evaluation through properties that make truthful probability assessment optimal.\"}]","A Review of Probabilistic Predictive Uncertainty Estimation with Machine Learning - Review | PDF",1785679300,209,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"a-review-of-probabilistic-predictive-uncertainty-estimation-with-machine-learning-review","",{"@graph":36,"@context":86},[37,54,69],{"@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/a-review-of-probabilistic-predictive-uncertainty-estimation-with-machine-learning-review/117741/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05","2026-08-02",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why should machine learning predictions be represented as probability distributions?","Question",{"text":76,"@type":77},"Probability distributions communicate uncertainty and increase the amount of information provided to end users compared with single point estimates.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does Bayesian modeling support predictive uncertainty estimation?",{"text":81,"@type":77},"Bayesian methods assign priors to model parameters, update them using data, and compute predictive uncertainty by integrating over the posterior distribution of parameters.",{"name":83,"@type":74,"acceptedAnswer":84},"What role do scoring functions and proper scoring rules play in evaluating probabilistic predictions?",{"text":85,"@type":77},"They provide principled metrics for assessing probabilistic forecasts, supporting consistent evaluation through properties that make truthful probability assessment optimal.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":46,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":46,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]