[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84403-en":3,"doc-seo-84403-105":29,"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":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},84403,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Algorithmic Expert Aggregation","Forecast aggregation combines multiple Bayesian experts’ predictions into a single aggregate forecast, but many existing approaches optimize a loss or robustness criterion without ensuring calibration to the realized outcome. The paper introduces expert aggregation, aiming to combine calibrated Bayesian experts into a new calibrated expert. The aggregator knows priors and which experts are available, but not the underlying Bayes probabilities. Results characterize which calibrated refinements are constructible and show efficient algorithms for randomized outputs, while deterministic refinement and proper-loss optimization are computationally intractable.","arXiv :2607 .08744v 1 [ cs .GT] 9 Jul 2026  \nAlgorithmic Expert Aggregation  \nWei Tang∗ Hanrui Zhang†  \nAbstract  \nForecast aggregation aims to combine information from multiple Bayesian experts’ forecasts into an aggregate forecast. In much of this literature, however, the aggregate forecast is optimized for a particular loss or robustness criterion and need not itself be calibrated with respect to the outcome: the reported forecast need not equal the conditional probability of the outcome given the aggregate forecast itself. We introduce and study expert aggregation, where the goal is instead to aggregate Bayesian experts into a new expert that continues to provide calibrated forecasts. In particular, we consider a setting where each input expert reports calibrated predictions, and the aggregator observes the prior distribution over states, and the input experts, but not the underlying Bayes probabilities of the states. We ask whether one can (i) construct a calibrated output expert that Blackwell refines a target expert and cannot be further Blackwell improved using the available information; and (ii) when a proper loss is specified, compute anearly loss-optimal expert among all such refinements.  \nWe formulate calibrated experts as reduced-form information structures and measure refinement by Blackwell dominance of the induced prediction distributions. We characterize the constructible output experts through observable linear information: the input experts generatea linear system whose row space determines which calibrated output predictions are identifiable, and a new expert is constructible exactly when its predictions lie in the associated observable nonnegative cone. We establish a sharp algorithmic picture. When randomized output experts are allowed, both the refinement-search question (i) and the proper-loss optimization question (ii) admit efficient algorithms. In contrast, deterministic output experts are computationally intractable: deciding whether a deterministic calibrated refinement exists is NP-hard even with two input experts where target expert is a constant base-rate expert, and deterministic properloss optimization admits no multiplicative PTAS unless P = NP, even for the Brier loss.  \n∗ Chinese University of Hong Kong. Email: [weitang@cuhk.edu.hk](weitang@cuhk.edu.hk)[ ](weitang@cuhk.edu.hk)†Chinese University of Hong Kong. Email: [hanrui@cse.cuhk.edu.hk](hanrui@cse.cuhk.edu.hk)  \n1 Introduction  \nA decision maker is debating whether to take a certain action whose payoff depends on whether a certain event happens: For concreteness, the payoff is 1 if the event happens, and −1 otherwise. To assist, a team of experts make forecasts on the probability that the event happens. Being professional, these experts always report their best-effort forecast, taking into consideration all the information they possess. Nonetheless, different experts specialize in different aspects of the matter, leading to different forecasts all being honest from the respective expert’s own point of view. The decision maker then faces a problem: How to aggregate these different forecasts into a single forecast that best informs the decision?  \nThis is the very research question addressed by the line of research on forecast aggregation (see, e.g., [Sto61 , BG69]) . In an idealized world where the decision maker has all the relevant prior knowledge, the above boils down to Bayesian inference: Given the joint information structure over the event concerned and all experts’ forecasts, it is (at least in principle) possible to infer the posterior probability of the event conditioning on all experts’ forecasts. On the other hand, oftentimes the decision maker cannot access the complete information structure (or form a belief about it in the Bayesian sense), in which case one may turn to robust forecast aggregation, which aims to optimize the aggregate forecast in the worst case over the uncertainty in the information structure [ABS18 , LR22","cbCaicyvaaej7aqW","https://ap.wps.com/l/cbCaicyvaaej7aqW","pdf",916319,1,54,"English","en",105,"# Abstract\n# Introduction\n## Problem setup and motivation\n## Expert aggregation and calibration requirement\n## Illustrative hospital/medical decision scenario","[{\"question\":\"What is the main goal of expert aggregation in this paper?\",\"answer\":\"To aggregate multiple Bayesian experts into a single output expert that continues to provide calibrated forecasts, rather than merely optimizing a loss or robustness criterion.\"},{\"question\":\"What information does the aggregator observe and what does it not observe?\",\"answer\":\"The aggregator observes the prior distribution over states and the input experts, but it does not observe the underlying Bayes probabilities of the states.\"},{\"question\":\"How do randomized and deterministic output experts differ computationally?\",\"answer\":\"With randomized output experts, both refining calibrated experts and proper-loss optimization admit efficient algorithms; with deterministic output experts, deciding existence of a calibrated refinement is NP-hard and proper-loss optimization has no multiplicative PTAS unless P=NP.\"}]",1784195341,136,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"algorithmic-expert-aggregation","",{"@graph":35,"@context":85},[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/algorithmic-expert-aggregation/84403/",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",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of expert aggregation in this paper?","Question",{"text":75,"@type":76},"To aggregate multiple Bayesian experts into a single output expert that continues to provide calibrated forecasts, rather than merely optimizing a loss or robustness criterion.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What information does the aggregator observe and what does it not observe?",{"text":80,"@type":76},"The aggregator observes the prior distribution over states and the input experts, but it does not observe the underlying Bayes probabilities of the states.",{"name":82,"@type":73,"acceptedAnswer":83},"How do randomized and deterministic output experts differ computationally?",{"text":84,"@type":76},"With randomized output experts, both refining calibrated experts and proper-loss optimization admit efficient algorithms; with deterministic output experts, deciding existence of a calibrated refinement is NP-hard and proper-loss optimization has no multiplicative PTAS unless P=NP.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"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":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]