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Howard and James E. Matheson presents a new form of problem description that bridges qualitative understanding and quantitative specification for decision analysis. The paper explains how influence diagrams represent relations, functions, and numerical values in both deterministic and probabilistic settings. It develops probabilistic dependence and independence concepts under a state of information, then derives probability expansion via the chain rule, noting that alternative expansions are logically equivalent yet differ in how easily decision makers can provide assessments.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/influence-diagrams-influence-diagrams-9-2005/665376/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/influence-diagrams-influence-diagrams-9-2005/665376.png","ImageObject",300,407,{"name":92,"@type":93},"ahduag","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does the influence diagram aim to solve in decision analysis?","Question",{"text":112,"@type":113},"It provides a representation that computers can treat formally while also being understandable to people across technical backgrounds, helping make uncertain information usable for decision making.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the paper distinguish probabilistic dependence from simple functional dependence?",{"text":117,"@type":113},"With uncertainty, a variable may be probabilistically dependent on others because learning specific values changes its probability, even if deterministic functional dependence does not apply.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the role of probabilistic independence in the representation?",{"text":121,"@type":113},"Independence is defined relative to an assessor’s state of information S, and it is expressed through equality relationships between assigned probability distributions.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665376,1793301840,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},3985747870414,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Downloaded from informs .org by [ [142.104.240.194](142.104.240.194)] on 25 April 2015, at 11:23 . For personal use only, all rights reserved.  \nDecision Analysis  \nVol. 2, No. 3, September 2005, pp. 127–143 issn 1545-8490 􀀁 eissn 1545-8504 􀀁 05 􀀁 0203 􀀁 0127  \ninforms ® doi 10. 1287/deca.1050.0020 © 2005 INFORMS  \nInﬂuence Diagrams  \nRonald A. Howard and James E. Matheson  \nStrategic Decisions Group  \nKey words: inﬂuence diagram; expansion; expansion order; decision tree; decision-tree order; Bayes; arrow reversal; decision network; decision-tree network; value of clairvoyance; Bayesian network; belief network; knowledge map  \n1. Introduction  \nThe rapid growth of electronic computation continues to challenge our ability to conceptualize and describe the world around us. Mathematical tools and formal descriptions serve poorly as a communication device with the majority of people not trained in nor used to mathematical means of expression. Yet virtually everyone has information useful in the solution of his own problems or the problems of others, if only it could be tapped.  \nThe subject of this paper is a new form of description, the inﬂuence diagram, that is at once both a formal description of the problem that can be treated by computers and a representation easily understood by people in all walks of life and degrees of technical proﬁciency. It thus forms a bridge between qualitative description and quantitative speciﬁcation.  \nThe reason for the power of this representation is that it can serve at the three levels of speciﬁcation of relation, function, and number, and in both deterministic and probabilistic cases. In the deterministic case, relation means that one variable can depend in a general way on several others; for example, proﬁt is a function of revenue and cost. At the level of function, we specify the relationship; namely, that proﬁt equals revenue minus cost. Finally, at the level of number, we specify the numerical values of revenue and cost and hence determine the numerical value of proﬁt.  \nIn the probabilistic case, at the level of relation we mean that given the information available, one variable is probabilistically dependent on certain other variables and probabilistically independent of still other variables. At the level of function, the probability distribution of each variable is assigned condi-  \ntioned on values of the variables on which it depends. Finally, at the level of number, unconditional distributions are assigned on all variables that do not depend on any other variable and hence determine all joint and marginal probability distributions.  \nAs an example of the probabilistic case, we might assert at the level of relation that income depends on age and education and that education depends on age. Next, at the level of function we would assign the conditional distribution of income given age and education and the distribution of education given age. Finally, at the level of number, we would assign the unconditional distribution on age.  \nThe successive degrees of speciﬁcation can be made by different individuals. Thus, an executive may know that sales depend in some way on price, but he may leave to others the probabilistic description of the relationship.  \nBecause of its generality, the inﬂuence diagram isan important tool not only for decision analysis, but for any formal description of relationship and thus for all modeling work.  \nIn the present paper, we shall focus on the probabilistic use of inﬂuence diagrams since the deterministic use is a special, but important, case of the probabilistic. We now proceed to development of the inﬂuence diagram concept, to examination of its properties, and to illustration of its use.  \n2. Probabilistic Dependence1  \nOne of the most perplexing aspects of making decisions under uncertainty is the problem of representing  \n1 This entire discussion applies as well to events as to variables.  \n127  \nDownloaded from informs .org by [ [142.104.240.194](1","cbCaibZOpe2FrUr5","https://ap.wps.com/l/cbCaibZOpe2FrUr5","pdf",305927,17,"English","# Introduction\n# Probabilistic Dependence\n# Probabilistic Independence\n# Expansion","[{\"question\":\"What problem does the influence diagram aim to solve in decision analysis?\",\"answer\":\"It provides a representation that computers can treat formally while also being understandable to people across technical backgrounds, helping make uncertain information usable for decision making.\"},{\"question\":\"How does the paper distinguish probabilistic dependence from simple functional dependence?\",\"answer\":\"With uncertainty, a variable may be probabilistically dependent on others because learning specific values changes its probability, even if deterministic functional dependence does not apply.\"},{\"question\":\"What is the role of probabilistic independence in the representation?\",\"answer\":\"Independence is defined relative to an assessor’s state of information S, and it is expressed through equality relationships between assigned probability distributions.\"}]","Inﬂuence Diagrams - Inﬂuence Diagrams 9 2005 | PDF",43]