421 Cost-Sharing in Bayesian Knowledge Bases Solomon Eyal Shimony and Carmel Domshlak Dept. ofath. andComp. ci. Eugene Santos Jr. Dept. ofElectrical andComp. Eng. Ben Guionniversity of the Nege .0.Bo653,Beer-heva84105,IAE e-mil:  shimony, carmel @cs.bgu.ac.il Abstract Bayesianowledgebases (BKBs) are agen ralization of Bayes networks and weighte proofgrahs (WAODAGs), thatallowcycles nthecausalgraph. easoning nBKBs e uires indin the most robable inferences onsistent with the vidence. The cost sharing heristic for iding least-cost ex planationsinWADAGswaspresented an shown to be eective by Charniak and Hu sain. owever, the cycles in BKBs would make te enition ocost-sharing cyclic s well, ifapplied directly to BKB. B teat ingthedening equations ofcost-sharingas asystem of equations, onecanroperly de ne an adissible cost-sharing heuistic for BKBs. Empiricalevaluationshowsthatcost sharng improves performance signcanty whenappliedtoBKBs. 1 INTRODUCTION Bayes networks 7] are a commonly used eaoning toolwithin the uncertaint in Acomunity. ately, graphcal causl probabiistic modes hae shown up that generalize on the acyclc Bayes networks, in or der to cater for causal phenomena whh cannot be strictly partialy ordered. These models have causal cycles 1, 8], or ndirected sections in the directed graphs 2NJ. Clearly, one sti! needs o do eie be liefrevisionorbeliefupdatng 7] inorder t perfor reasoniginthesechemes.These moregeneral mod els, being less restrictive, pause ineresting problems ileentngeoningagorithmsfor them. Baysian knowledge ase (BKBs) [8] ar a gneral ization of Bays networks and weighted (AND/O, dreted acycli) proof gaphs (acronym WAODAGs) 4,that allow cclesint causalgraph. Considerthe prolem of inding the mst probablenference ("ex planation") consisten with the evidene in a BKB. Thisproblemisanalogousto(andmregeneralthan) theN-hardproblemofbeliefreision inBaysnet- AirForcenstitute ofechnlogy Wright-Patterson AFB, OH e-ail:esantos@ait.af.mil woks,ordnginimum-costpoof onaWAODAG. As for Bayes networks, reasoning with tre-shaped BKBscanbedoneecently. However,i tislearthat inactualappicationswecannotusualyforcourrep esentation tobelongtotheeasyclassofprbems. To-date, inding most-probable inference in general BKBs h been implemented s bes-ist heursc search, whee the heuristic used wascost-so-far, with dismal results. The reason is that this lcal heuris tic does no take into account the cost of odes (or variables) tbeassignedateronnthesearch. Prop agation ofcoststobeincurredismuchpreferable,but it isnon-trivial to do s oinamanner resting in an ssleeurstc. The latter wasrstachievedby usingthecost-sharingpropagatedcostmethd3(see nextsectio forabrief denition). twasshownbyCharniakandHusain [3] thatforind igest-ct epnationsnWAODAGs,the(admis ible)cost-sharingheuristichas amuchbetterperfor mance. Thecostsharingheursti wasalsofounduse fulfor beliefrevision inBayesnetworks 9). Here, we generalizecost-sharing to apply tocyclic graphs, and show that the resultingheristc i asoadmissble. Thegeneralization ofthecost-sharing heuristic, while straightforward, causes several problems. irst, the cyclesntheBKBmaketheproblemofproperly dein ing the heuristic nontrivial. If weustsed the same deiningequations,thefact thatthereaecyleswould make the dening equations cyclic. But by looking a theseequations as a system ofequations, we state hat a io to th sse is o heris. An suchsoutin to the systemofeqations is shownto be an admssible heuristic. A secnd problem ishow tosoletheseequations. Thestandardtop-downago ithmused inpio wo wouldbehnderedbythecy cles: eveniwe conert i toakindofmessage-passing updangalgorithm, in manycases the algorithmwil oopindeinitely. nstead,weshowthatconertingthe ystemofsemi-linearequationstoainearprogram, we nvalt theheristi n polynomial time. Webegin with a motivatigBB eamp, folowed by a formal deniton of BKBs (secton 2). We then reae BKBs toWAODAs,andreviewthecostsar-