[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125192-en":3,"doc-seo-125192-105":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},125192,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","A Study of Jeffrey’s Rule With Imprecise Probability Models","Jeffrey’s rule provides a method for updating beliefs about a probability measure when new information is given conditionally on a partition of the possibility space, while preserving the original marginal information on the partition. The paper connects this update rule to total probability and to marginal extension results for coherent lower previsions, then generalizes the formulation to other imprecise models, including choice functions, possibility measures, and distortion models.","A Study of Jeffrey’s Rule With Imprecise Probability Models  \nEnrique Miranda [mirandaenrique@uniovi.es](mirandaenrique@uniovi.es)  \nStatistics and Operations Research, University of Oviedo, Spain  \nArthur Van Camp [arthur.vancamp@bristol.ac.uk](arthur.vancamp@bristol.ac.uk)  \nDepartment of Philosophy, University of Bristol, UK  \nAbstract  \nJeffrey’s rule tells us how to update our beliefs about a probability measure when we have updated information conditional on some partition of the possibility space, while keeping the original marginal information on this partition. It is linked to the law of total probability, and is therefore connected to the notion of marginal extension of coherent lower previsions. In this paper, we investigate its formulation for some other imprecise probability models that are either more general (choice functions) or more particular (possibility measures, distortion models) than coherent lower previsions.  \nKeywords: Jeffrey’s rule, marginal extension, coherent lower previsions, sets of desirable gambles, nonadditive measures, choice functions  \n1. Introduction  \nConsider a finite possibility space 􀁓, and a partition B of 􀁓 . Given a probability measure 􀀥 on P (􀁓) , 1 it is possible to relate the probability 􀀥( 􀀖) of any event 􀀖 to the probabilities 􀀥(􀀗) of the events 􀀗 in the partition B and the probabilities 􀀥( 􀀖|􀀗) conditional on events 􀀗 in B ,2 using the law of total probability:  \n􀀥 ( 􀀖) = ∑︁ 􀀥 (􀀗)􀀥 ( 􀀖|􀀗) .  \n􀀗 ∈B  \nSuppose that we “observe” a new ‘input’ probability measq  \nure 􀀥 on B. If we now want to obtain a new probability (s()|l()thfo􀀖of|ta)tsllfaisfiinalrobles tB􀀗ab;heinilitcByonstrandai􀀖n[,ein[g onrigidiBty]] ( 􀀖) = 􀀗 ( 􀀖|􀀗)(􀀗) = 􀀗 􀀥 ( 􀀖(|􀀗Jeff) e(􀀗y’s) rule)  \n1We let P ( X) be the power set of its input set X. Elements of P (􀁓)  \nare called events.  \n2We assume in this introductory section, for simplicity, that 􀀥 (􀀗) > 0 for every 􀀗 in B, making sure that 􀀥 ( 􀀖|􀀗) is well defined.  \nfor all 􀀖 ⊆ 􀁓, so the two constraints above are a unique descrThipetionequiovfalt[ 18exp, 19ect]a. tion operator version of Jeffrey’s  \nrule is given by  \nwhere L(􀀵) i)s =ths(􀀚eto(􀀵f a|Bll()Jr)eeffal-rfveoyarl’asullreleminfaopLrseo(􀁓)ec—tationcallesd) gambles—and sometimes denoted by L when it is clear from the context what the domain 􀁓 is.  \nIn this paper, we investigate the formulation of Jeffrey’s rule in the context of imprecise probabilities. One prominent such connection was already established by Peter Walley in his celebrated marginal extension theorem in [41] . He showed that the law of total probability can be extended to coherent lower previsions in the following manner: given a coherent lower prevision 􀀥  B on the class of Bmeasurable gambles and a separately coherent conditional lower prevision 􀀥 ( ·|B) on L(􀁓), the smallest coherent lower prevision that is coherent with 􀀥  B , 􀀥 ( ·|B) is given by  \n􀀥 B 􀀥  B (􀀥 ( ·|B)) .  \nThis result was later extended to a finite number of conditional lower previsions in [24] and to sets of desirable gambles in [10, Thm. 3] . In this paper, we shall look atthe formulation of this result for other different families of imprecise probability models, be it more general or more specific ones than coherent lower previsions. Given one such family C, we consider thus a marginal and a conditional model within this family, and look for a global model that  \n(a) belongs to the same class C of uncertainty models asthe marginal and conditional, and  \n(b) is compatible with them in the manner we shall specify later on.  \nIn case there exists such as model, we may then analyse whether this model is unique, or, if it is not, whether it is possible to characterise the smallest such model. This would be a sort of natural extension, with the additional assumption of compatibility (that perhaps may be stronger  \n© 2023 E. Miranda & A. Van Camp.  \nA Study of Jeffrey’s Rule With Imprecise Probability Models  \nthan usual coherence) and the structural assumption of being an el","cbCait8wwMXmitdB","https://ap.wps.com/l/cbCait8wwMXmitdB","pdf",352619,1,12,"English","en",105,"# Introduction\n## Jeffrey’s Rule for Sets of Desirable Gambles","[{\"question\":\"What does Jeffrey’s rule update and what information does it preserve?\",\"answer\":\"It updates beliefs about a probability measure using information conditional on a partition, while preserving the original marginal information on that partition.\"},{\"question\":\"How is Jeffrey’s rule linked to total probability in this paper?\",\"answer\":\"The paper notes that Jeffrey’s rule is linked to the law of total probability, connecting it to marginal extension ideas for coherent lower previsions.\"},{\"question\":\"Which imprecise probability models are studied beyond coherent lower previsions?\",\"answer\":\"The paper investigates formulations for broader or more specific imprecise frameworks such as choice functions, possibility measures, and distortion models.\"}]","A Study of Jeffrey’s Rule With Imprecise Probability Models | 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does Jeffrey’s rule update and what information does it preserve?","Question",{"text":75,"@type":76},"It updates beliefs about a probability measure using information conditional on a partition, while preserving the original marginal information on that partition.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is Jeffrey’s rule linked to total probability in this paper?",{"text":80,"@type":76},"The paper notes that Jeffrey’s rule is linked to the law of total probability, connecting it to marginal extension ideas for coherent lower previsions.",{"name":82,"@type":73,"acceptedAnswer":83},"Which imprecise probability models are studied beyond coherent lower previsions?",{"text":84,"@type":76},"The paper investigates formulations for broader or more specific imprecise frameworks such as choice functions, possibility measures, and distortion 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