EVIDENCE ABSORPTION AND PROPAGATION THROUGH EVIDENCE REVERSALS Ross . Shachter DepartmentofEngineering-Economic Systems Stanford Univeity Stanford, CA 94305-4025 shachter@sumex-aim.stanford.edu The arc reversanode reduction approach to probabiliscinfernceisextended to include the case of instantiated evidence by an operation caed "evidence reversal." This not only provides a techniue for computing posterior j oin distributions on general belief network, butalso provides insight into the methods of Pearl 1986a] and Lauritzen and Spiegelhalter 1988]. Althoughit is wellunderstood that te latter two agorithms are closely related,in fact all three algoriths are identical whenever the blief netwo isa forest. Q. Introduction recent ears,themain clases ofexact algorithms have emergedosolveprobabilistic infernceproblemsfonnuladasblief networks (orinflence diagrams). The propagaon method ofPeal 1986a] is a efficient,message-passing approachfor pe  (singly-connected networs,ordrctedgraphsinwhich there are no undirected cycles),whichcanbgeneralied throughconditioning tomor generanetworks [Pearl 1986b]. Themethodofarc vealsand nodereductions [Shachter 1986, 1988] processes generanetworsthrough tological transfonnationswhich presee criteriavalues and j oint distributions. Thenewestmethodwas deveoped by Lauriten and Spiegelhalter [ 1988] andgeneraliedtoDempster-Shafer belief fnctions by Dempsterand Kong [1986]and Shafer  1987 ] . Itconstructs a chorda undirected graph analog to the blief networkin rder to obtain processingefficiencyon general networkssimilartoPearl'smethod. Itworksona spcial case of a pltre,a f(a disconnected setoftrees,oradirected acycic graphinwhichno nodehasmor thanoneparnt). Inthispaper, the methodofarcreverals and node edctions isextendedto efcientlyhandle exprimentalevidence. Originallydeveloped for pcessingof infuence diagrams withdecisions (Howard and Matheon 1981], ths esal appachusessilegraphreductionso transfo the topoogyof the networ, hile maintainingtheoint distribtionof a subset of the variablesorthe (excted) value ofcrπterion variables [Osted 198, hahtr 1986, 1988]. Indecisionanalysiswithseuena decisions, mostof te exprimentaevidencisobseed afertheinialdecision,sothe emphasisinthe methodhasbenonpr-steoranalysis,thatis, planningfor  pssiblevaluesofthe exprimenaloutcome. Inthispap,however, spciacaristakentoefcientlyprcessevidence whichisobed priort theinitialdecision. On adiagramwithoutanydecisions,thi sisprcisely thepbabilisticinferenceproblemonblief networks. Secon Â defnesthe diagra while Section 3 defnesaevidencene and the oprationsof evidenc aoron, veal, ad ppaaton. Section 4 intrducesprbabilitprpagation, and thecontrlof theseoprationsisdescribdin Section ✆. Section6 contrasts thismetodwith thoseofPearlandofLauritzen  Spiegehalter andSecon  conains me conclusions. 2. Belief Diagrams Tenecessarynotaonispresentedinthissection todefƣnea the blefdiaga, ageneralizationof a probabilisticinfencediagram. Althoughthe resultsinthispaprcaneasily b appliedtothe general infuencediagmwithdecision and value nodes,onlychance nodes,rprsenting random variables, will b used simplify thepresentation. Thisprobabiisc infuence diagra wihevidence nodes correspnds exactlyto bliefneworks, and frm heron,theywill brfeedtoasblief diagrams. A blie dia is anetworbuiltonadircted acyclicgraph. The babilistic nodesN= 1, . .. , n correspnd to randomvariables X = , .. . • X n . EachvariableXjhas asetpssibleoutcomes, j, and a conditionaprobability disibution,1j, over those outcomes. The conditioning variablesfor1j have indces in the set ofparentsor conditinal ps, C(),C()N, ad ar indicatedin thegraphby arcs from thenodes in C() into node j . If njisamarginaldistribution, then C()isthe emptyset, 0. Eachprobablisticvariable Xjisinitially unobserved, butat sometimeitsvauexj e ' mighbcome known. Atthatpint, it bcomes an evidencevariabe,asdiscussedinthenextsection. As aconventio,a lowercaseletter rpresentsa singlenode inthegraphand an uprcaseletter epsen asetofnodes. If isasetofnodes, . thenX G denotes thevectorof variables 0