Eugene Charniak and Eugene Santos Jr. Department of Computer Science Brown University Providence RI and

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1 From: AAAI-92 Procdings. Copyright 1992, AAAI ( All rights rsrvd. mic MAP Calcul Eugn Charniak and Eugn Santos Jr. Dpartmnt of Computr Scinc Brown Univrsity Providnc RI and Abstract W prsnt a dynamic algorithm for MAP calculations. Th algorithm is basd upon Santos s tchniqu (Santos 1991b) of transforming minimal-cost-proof problms into linarprogramming problms. Th algorithm is dynamic in th sns that it is abl to us th rsults from an arlir, nar by, problm to lssn its sarch tim. Rsults ar prsntd which clarly suggst that this is a powrful tchniqu for dynamic abduction problms. Introduction Many AI problms, such as languag undrstanding, vision, mdical diagnosis, map larning tc. can b charactrizd as abduction - rasoning from ffcts to causs. Typically, whn w try to charactriz th computational problm involvd in abduction, w distinguish btwn (a) finding possibl causs and (b) slcting th on most likly to b corrct. In this papr w concntrat on th scond of ths issus. W furthr limit ourslvs to probabilistic mthods. It sms clar that th most likly to b corrct is that which is most probabl. Th argumnts against th us of probability thrfor usually concntrat on th difficulty of computing th rlvant probabilitis. Howvr, rcnt advancs in th application of probability thory to AI problms, from Markov Random Filds (Gman & Gman 1984) to Baysian Ntworks (Parl 1988) hav mad such argumnts hardr to maintain. Th problms of our first sntnc ar not just abduction problms, thy ar dynamic abduction problms. That is, w ar not confrontd by a singl problm, but rathr by a squnc of closly rlatd problms. In mdical diagnosis th problm changs as nw vidnc coms in. In vision, th pictur changs slightly ovr tim. In story undrstanding, w gt furthr portions of th story. Of cours w can rduc th dynamic cas to th static. Each scn, mdical problm, or story comprhnsion task sts up a nw probabilistic problm which th program tackls afrsh. For xampl, in our prvious work on th Wimp3 story undrstanding program (Charniak & Goldman 1989; Charniak & Goldman 1991; Goldman & Charniak 1990) story dcisions ar translatd into Baysian Ntworks. Th ntwork is xtndd on a word-by-word basis. As on would xpct, most of th ntwork at word n was thr for word n - 1. Yt th algorithm w usd for calculating th probabilitis at n mad no us of th valus found at n - 1. This smd wastful, but w knw of no way to do bttr. This papr corrcts this situation. Prvious Rsarch Roughly spaking, probabilistic algorithms can b dividd up into xact algorithms and ons which ar only likly to rturn a numbr within som 6 of th answr. Considring xact algorithms, w ar not awar of any which ar dynamic. Th story is diffrnt for th approximation schms sinc most ar stochastic in natur and hav th proprty that if w can start with a bttr guss of th corrct answr thy will convrg to that answr mor quickly. (S for xampl (Shachtr & Pot 1989; Shw & C,oopr 1990).) Thus such algorithms ar inhrntly dynamic in th sns that th rsults from th prvious computation can b usd to improv th prformanc on th nxt on. Howvr, vn for such algorithms w ar unawar of any study of thir dynamic proprtis. Th particular (xact) algorithm w propos hr did not aris from a sarch for a dynamic algorithm, but rathr from an ffort to improv th spd of probabilistic calculations in Wimp3. On schm w considrd was th us of Hobbs and Stickl s (Hobbs t al. 1988) work on minimal-cost proofs for finding th bst xplanations for txt. Thir ida was to find a st of facts which would allow th program to prov th contnts of th txtual input. Sinc thr would b many possibl sts of xplanations, thy assignd costs to assumptions. Th cost of a proof was th sum of th cost of th assumptions it usd. (Actually, w ar outlining cost-basd abduction as dfind in (Charniak 8z Shimony 1990) which is slightly simplr than th schm usd by Hobbs and Stickl. Howvr, th two ar sufficintly clos that w will not bothr to distinguish 552 Rprsntation and Rasoning: Abduction and Diagnosis

2 4 6 3 Figur 1: An and-or dag showing svral proofs of hous-quit btwn thm.) For xampl, s Figur 1. Thr w want th minimal-cost proof of th proposition hous-quit (my hous is quit). This is an and-nod so proving it rquirs proving both no-barking and TV-off. Figur 1 shows svral possibl proofs of ths using diffrnt assumptions. On possibl st of assumptions would b homwork-tim, dog-slping. Givn th costs in th figur, th minimal cost proof would b simply assuming kids-walking-dog at a cost of 6. W wr intrstd in minimal-cost proofs sinc thir math sms a lot simplr than typical Baysian Ntwork calculations. Unfortunatly, th costs wr adhoc, and it was not clar what th algorithm was rally computin. Charniak and Shimony (Charniak & Shimony 1990 j fixd ths problms by showing how th costs could b intrprtd as ngativ log probabilitis, and that undr this intrprtation minimalcost proofs producd MAP assignmnts (subjct to som minor qualifications). (A MAP, Maximum A- Postriori, assignmnt is th assignmnt of valus to th random variabls which is most probabl givn th vidnc.) A subsqunt papr (Shimony & Charniak 1990), showd that any MAP problm for Baysian Ntworks could b rcast as a minimal-cost-proof problm. In th rst of this papr w will talk about algorithms for minimal-cost proofs and dpnd on th rsult of (Shimony & Charniak 1990) to rlat thm to MAP assignmnts. Unfortunatly, th bst-first sarch schm for finding minimal-cost proofs proposd by Charniak and Shimony (and modld aftr th Hobbs and Stickl s work) did not sm to b as fficint as th bst algorithms for Baysian Ntwork valuation (Jnsn t al. 1989; Lauritzn & Spiglhaltr 1988). Evn improvd admissibl cost stimations (Charniak dz Husain 1991) did not rais th MAP prformanc to th lvls obtaind for Baysian Ntwork valuation. Rcntly, Santos (Santos 1991a; Santos 1991c; Santos b) prsntd a nw tchniqu diffrnt from th bst-first sarch schms. It showd how minimal-cost proof problms could b translatd into O-l programming problms and thn solvd using simplx combind with branch and bound tchniqus whn simplx did not rturn a O-l solution. Santos rports that simplx rturns O-l solutions on 95% of th random waodags h trid and 60-70% of thos gnratd by th Wimp 3 natural languag undrstanding program. This plus th fact that branch and bound quickly found O-l solutions in th rmaining cass showd that this nw approach outprformd th bst-first huristics. It actually xhibitd an xpctd polynomial run-tim growth rat whr as th huristics wr xponntial. This approach immdiatly suggstd a dynamic MAP calculation algorithm. Waodags as Linar Constraints Santos, following (Charniak & Shimony 1990) formalizd th minimal-cost proof problm as on of finding a minimal-cost labling of a wightd-and-or-dag (waodag). Informally a waodag is a proof tr (actually a dag sinc a nod can appar in svral proofs, or in th sam proof svral tims). W hav alrady sn an xampl in Figur 1. Formally it is a 4-tupl< G, c, T, S >, whr 1. G is a connctd dirctd acyclic graph, G = (V, E), 2. c is a function from V to th non-ngativ rals, calld th cost function (this is simplifid in that w only allow assuming things tru to hav a cost), 3. r is a function from V to (and, or) which labls ach nod as an and-nod or an or-nod, 4. S is a st of nods with outdgr 0, calld th vidnc. Th problm is to find an assignmnt of tru and fals to vry nod, obying th obvious ruls for and-nods and or-nods, such that th vidnc is all labld tru, and th total cost of th assignmnt is minimal. In th transformation to a linar programming problm a nod n bcoms a variabl 3: with valus rstrictd to O/l (fals/tru). An and-nod n which was Charniak and Santos 553

3 tru iff its parnts no... ni wr tru would hav th linar constraints AND1 x 5 x0,...x 5 zi AND2 x > x xi - i + 1. An or-nod n which is tru iff at last on of its parnts no... ni ar tru would hav th linar constraints OR1 OR2 x * * * + xi x 1 x0,...x 2 xi. Th constraints AND1 and OR1 can b thought of as bottom up in that if on thinks of th vidnc as at th bottom of th dag, thn ths ruls pass tru valus up th dag to th assumptions. Th constraints AND2 and OR2 ar top down. In our implmntation w only bothrd with th bottom-up ruls. Th costs of assumptions ar rflctd in th objctiv function of th linar systm. As w ar making th simplifying (but asily rlaxd) assumption that w assign costs only to assuming statmnts tru, modling th costs co... ci for nods no... n; is accomplishd with th objctiv function coxo * + cixi Santos thn shows that a zro-on solution to th linar quations which minimizs th objctiv function must b th minimal-cost proof (and thus th MAP solution for th corrsponding Baysian Ntwork.) Th Algorithm Simplx is, of cours, a sarch algorithm. It sarchs th boundary points of a convx polygon in a high dimnsional spac, looking for a point which givs th optimal solution to th linar inqualitis. Th sarch starts with an asily obtaind, but not vry good, solution to th problm. It is wll known that simplx works bttr if th initial solution is closr to optimal. Whn w say bttr w man that th algorithm rquirs fwr pivots - th traditional masur of tim for simplx. To thos not familiar with th schm, th numbr of pivots corrsponds to th numbr of solutions tstd bfor th optimal is found. In what follows w will assum that th following ar primitiv oprations, and that it is possibl to tak an xisting solution, prform any combination of thm, and still us th xisting solution as a start in finding th nxt solution. This is, in fact, standard in th linar-programming litratur (s (Hillir & Librman 1967)). Ll Add a nw variabl (with all zro cofficints) to th problm. L2 Add a nw quation to th problm. L3 Chang th cofficint of a variabl in an quation. L4 Chang th cofficint of a variabl in th objctiv function. Moving down a lvl, w nd to talk about oprations on th waodags. In particular w nd to do th following: Wl add a nw nod to th dag, W2 add an arrow btwn nods, W3 W4 st th valu of a nod, rmov a nod (and arcs from it). For xampl, whn w gt a nw word of th txt (.g., bank ) in a story comprhnsion program w nd to add a nw or-nod corrsponding to it (Wl) and, sinc this is not a hypothsis, but rathr a known fact, st th valu of th nod to tru (W3). To add an xplanation for this word, say that it was usd bcaus th author wantd to rfr to a savings institution, w would add a scond or-nod corrsponding to th savings institution hypothsis, add a cofficint corrsponding to its cost to th objctiv function (L4), and add an arc btwn it an th nod for th word (W2), possibly with an intrmdiat nwly mintd and-nod. Nxt w rlat Wl-W4 to Ll-L4. Wl-W3 ar fairly simpl. Wl Adding a nw nod nl to th dag is accomplishd by adding a nw variabl xi to th linar programming problm. If it has a cost cl, chang th cofficint of xi in th objctiv function from 0 to cl u-4 W2 Adding an arrow from nod nr to n2 is accomplishd in on of thr ways (w only dal with th bottom-up quations): If n2 is an and-nod, add th nw quation 22 5 Xl p-4 If n2 is an or nod with th quation 22 5 x xi alrady in th problm, chang th cofficint of xi in this quation from zro to on (L3) o Els, add a nw quation x2 5 x1 (L2). W3 Stting th valu of a nod ni to tru (fals) is accomplishd by adding th quation nl = l(0) to th linar programming problm (L2). (In cass wr th nod is about to b introducd, w can optimiz by not xplicitly adding its variabl to any quations, and instad substituting its valu into th quations for parnts nods.) Th only complicatd modification to Woadags is W4, rmoving a nod (and arcs from it). W nd this so, as th txt gos along, it will b possibl to st valus of nods to b dfinitly tru or fals, thus rmoving thm from furthr probabilistic calculations (xcpt should thy srv as vidnc for othr facts). If all w wantd to do was st th valu, w could us th mthod in (W3), adding an quation stting th valu. But this is not nough. W want to rmov th variabl from all quations as wll so as to rduc th siz of th simplx tablau and thus dcras th pivot tim. W will assum in what follows that th variabl in qustion is to b assignd th valu it has in th currnt MAP. If this is not th cas thn a nw quation must b addd. 554 Rprsntation and Rasoning: Abduction and Diagnosis

4 If th variabl xj to b rmovd is a non-basic variabl th problm is asy. W just us tchniqu L3 to modify all of zi s cofficints to b zro. W can thn rmov its column from th tablau. This dos not work, howvr, if th variabl is basic. Thn thr will b a row, say ri which will look lik this: Ci,l,Ci,2, *'* ci,j-1, k,j+1, ***Ci,k = b. Changing th cofficint ci,j which is currntly 1, to 0 will lav this quation with no basic variabl, somthing not allowd. Howvr, if any of th othr cofficints ci,l... ci,k 2 0 thn w can pivot on that variabl and mak it basic. This lavs th cas whn non of th othr variabls ar positiv. Th solution dpnds on th fact that w ar stting th valu of x~j to its currnt valu in th MAP. This will b, of cours, b,. Thus w should, and will, subtract b, from both sids of th quation. On th lft-hand-sid this is don by making ci,j = 0. On th right w actually subtract b, - b, to gt b: = 0. Not now that sinc bi = 0 w can multiply both sids of th quation by -1 whn all of th othr cofficints ar ngativ, thus making thm positiv. b: is still zro so th quation rmains in standard form, and w now hav a non-zro cofficint to pivot on. Finally, if thr ar no non-zro cofficints othr than ci,j thn th quation can b dltd, along with XC;,~. sults Th algorithm dscribd in th prvious sction was run on a st of probabilistic problms gnratd by th Wimp 3 natural-languag undrstanding program. As w hav alrady notd, Wimp 3 works by translating problms in languag undrstanding into Baysian Ntworks. Ambiguitis such as altrnativ rfrncs for a pronoun bcom sparat nods in th ntwork. Th altrnativ with th highst probability givn th vidnc is slctd as th corrct intrprtation. Wimp 3 usd a vrsion of Lauritzn and Spiglhaltr s algorithm (Lauritzn & Spiglhaltr 1988) as improvd by Jnsn (Jnsn t ab. 1989), to comput th probabilitis. Wimp 3 gnratd 218 ntworks ranging in siz from 3 to 87 nods. As bfits an inhrntly xponntial algorithm, w hav plottd th log of valuation tim against ntwork siz. (All tims ar compild Lisp run on a Sun Spar 1.) S Figur 2. A last squars fit of th quation tim = B. 10A Idagl givs A =.03 and B =.275. Th total valuation tim for all of th ntworks was 575 sconds, to b compard with a total of 1235 sconds to run th xampls (or 46.5% of th total running tim). This illustrats that ntwork valuation tim was th major componnt of th tim takn to procss th xampls. Wimp 3 was thn modifid slightly to us th dynamic MAP algorithm dscribd in this papr. Whil thr is a gnral algorithm for turning Baysian ntworks into cost-basd abduction problms (Shimony & Charniak 1990), it can b rathr xpnsiv in trms of th numbr of nods cratd. Instad w mad us of th fact that th ntworks cratd by Wimp wr alrady prtty much and-or dags, and fixd th fw rmaining placs on a cas-by-cas basis. A graph of th rsulting programs running tim (or th log throf) vs. dag siz is givn in Figur 3. Th dag s sizs rportd ar, in gnral twic thos for th Baysian Ntworks. Th major rason is that th dag s includd xplicit and-nods whil ths wr absorbd into th distributions in th ntworks. Th total tim to procss th dags was 73 sconds. Thus th tim spnt on probabilistic calculations was dcrasd by a factor of 7.9. Running tim, of cours, is not a vry good masur of prformanc sinc so many factors ar blndd into th final rsult. Unfortunatly, th schms oprat on such diffrnt principls that w hav not bn abl to think of a mor implmntation indpndnt masur for comparing thm. It is, nvrthlss, our blif that ths numbrs ar indicativ of th rlativ fficincy of th two schms on th task at hand. As should b clar from Figur 3, th connction btwn th siz of th dag and th running tim is much mor tnuous for th linar tchniqu. This suggstd looking at th standard masur for th prformanc of th simplx mthod - numbr of pivots. Figur 4 shows th numbr of pivots vs DAG siz. Not that w did not graph dag siz against th Iog of th numbr of pivots. As should b clar, th numbr of pivots grows vry slowly with th siz of th dag, and th corrlation is not vry good. It crtainly dos not sm to b growing xponntially. If on dos a last squars fit of th linar rlation pivots = A- 1 dag 1 +B on finds th bst fit A =.091 and B = 7.1. Sinc th tim pr pivot for simplx is ordr n2 w hav smingly rducd th xpctd tim complxity of our problm to low polynomial tim. Conclusion Whil th rsults of th prvious sction ar imprssiv to our ys, it should b mphasizd that thr ar limitations to this approach. Th most obvious is that vn an n2 algorithm is only fasibl for smallr n. Cur algorithm prforms accptably for our ntworks of up to 175 nods. It is doubtful that it would work for problms in pixl-lvl vision, whr thr ar, say lo6 pixls, ach a random variabl. Furthrmor, if th problm is not amnabl to a cost-basd abduction formulation it is unlikly that th gnral transformation from Baysian Ntworks to waodags will produc accptably small dags. But this still lavs at lot of problms for which this tchniqu is applicabl. In svral prvious paprs (Charniak 1991; Charniak & Goldman 1991) on th us of Baysian Ntworks for story undrstanding w hav mphasizd that th problm of ntwork valuation was th major stumbling block in th us of ths ntworks. Th rsults of th prvious sction would indicat that this is no longr th cas. Putting th Charniak and Santos 555

5 l c d0 40 d0 d0 Figur 2: Baysian-Ntwork-Evaluation Tim (in Sc.) vs. Numbr of Nods l.Ol I :. 0. : 00 a. a 0 8 I I I I I I I I $5 d0 15 lb0 155 I Figur 3: Linar Cost-Basd Abduction Tim (in Sc.) vs. Numbr of Nods 6i 0 8 :a * 00 6 o- 8 Figur 4: Numbr of Pivots vs. Numbr of Nods 556 Rprsntation and Rasoning: Abduction and Diagnosis

6 data in prspctiv, w ar spnding about.3 sconds pr word on probabilistic calculations. Furthrmor, w ar using a crud vrsion of simplx which on of th authors wrot. It includs non of th mor sophisticatd work which has bn don to handl largr linar programming problms. It sms clar that a bttr simplx could handl ordr of magnitud largr problms in th sam tim (or lss, with th vry much mor powrful machins alrady on th markt.) Thus ntwork valuation is no longr a limiting factor. Now th limiting factor is for us, as it is for prtty much th rst of traditional AI, knowldg rprsntation. Acknowldgmnts This rsarch was supportd in part by NSF contract IRI and QNR contract N J frncs Charniak, E. and Goldman, R A smantics for probabilistic quantifir-fr first-ordr languags, with particular application to story undrstanding. In Procdings of th IJCAI Confrnc. Charniak, E. and Goldman, R A probabilistic modl of plan rcognition. In Procdings of th AAAI Confrnc. Charniak, E. and Husain, S A nw admissibl huristic for minimal-cost proofs. In Procdings of th AAAI Confrnc. Charniak, E. and Shimony, S. E Probabilistic smantics for cost basd abduction. In Procdings of th AAAI Confrnc. 106-l 11. Charniak, E Baysian ntworks without tars. AI Magazin 12(4): Gman, S. and Gman, D Stochastic rlaxation, gibbs distribution, and th baysian rstoration of imags. IEEE Transactions on Pattrn Analysis and Machin Intllignc 6~ Goldman, R. and Charniak, E Dynamic construction of blif ntworks. In Procdings of th Confrnc on Uncrtainty in Artificial Intllignc. Hillir, F. S. and Librman, G. J Introduction to Oprations Rsarch. Holdn-Day, Inc. Hobbs, J. R.; Stickl, M.; Martin, P.; and Edwards, D Intrprtation as abduction. In Procdings of th 26th Annual Mting of th Association for Computational Linguistics. Jnsn, F. V.; Lauritzn, S. L.; and Olsn, K. G Baysian updating in rcursiv graphical modls by local computations. Tchnical Rport Rport R 89-15, Institut for Elctronic Systms, Dpartmnt of Mathmatics and Computr Scinc, Univrsity of Aalborg, Dnmark. Lauritzn, S. L. and Spiglhaltr, D. J Local computations with probabilitis on graphical structurs and thir applications to xprt systms. J. Royal Statistical Socity 50(2): Parl, J Probabilistic Rasoning in Intllignt Systms: Ntworks of Plausibl Infrnc. Morgan Kaufmann, San Mato, CA. Santos, E. Jr. 1991a. Cost-basd abduction and linar constraint satisfaction. Tchnical Rport CS-91-13, Dpartmnt of Computr Scinc, Brown Univrsity. Santos, E. Jr. 1991b. A linar constraint satisfaction approach to cost-basd abduction. Submittd to Artificial Intllignc Journal. Santos, E. Jr. 1991c. On th gnration of altrnativ xplanations with implications for blif rvision. In Procdings of th Confrnc on Uncrtainty in Artificial Intllignc. Shachtr, R. D. and Pot, M. A Simulation approachs to gnral probabilistic infrnc on blif ntworks. In Procdings of th Confrnc on Uncrtainty in Artificial Intllignc. Shimony, S. E. and Charniak, E A nw algorithm for finding map assignmnts to blif ntworks. In Procdings of th Confrnc on Uncrtainty in Artificial Intllignc. Shw, M. and Coopr, G An mprical analysis of liklihood-wighting simulation on a larg multiplyconnctd blif ntwork. In Procdings of th Confrnc on Uncrtainty in Artificial Intllignc. Charniak and Santos 557