Scaling Up POMDPs for Dialog Management: The Summary POMDP Method. Jason D. Williams and Steve Young

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1 Scalng Up POMDPs for Dalog Managemen: The Summary POMDP Mehod Jason D. Wllams and Seve Young Cambrdge Unversy Engneerng Deparmen Trumpngon Sree, Cambrdge CB2 1PZ, UK BSTRCT Parally Observable Markov Decson Processes (POMDPs have been shown o be a promsng framework for dalog managemen n spoken dalog sysems. However, o dae, POMDPs have been lmed o arfcally small asks. In hs work, we presen a novel mehod called a Summary POMDP for scalng -fllng POMDP-based dalog managers o cope wh asks of a realsc sze. n example dalog problem ncorporang a user model bul from real dalog daa s presened. dalog manager s creaed usng hs mehod and evaluaed usng a second user model creaed from held-ou dalog daa. Resuls confrm ha summary POMDP polces scale well, and also show ha summary POMDP polces are reasonably robus o varaons n user behavor. 1. INTRODUCTION Dalog managemen for spoken dalog sysems can be approached as plannng under uncerany. (Fully-observable Markov decson processes ((FOMDPs and parally observable Markov decson processes (POMDPs offer a prncpled framework n hs pursu [5]. MDPs offer an accessble body of opmzaon echnques and have been successfully used o creae dalog managers for problems of realsc szes [4, 6, 9, 10]. However, MDPs do no ake proper accoun of he corrupon nroduced by he speech recognon channel. The POMDP framework, whch s desgned o cope wh nosy npu, has been found o conssenly ouperform MDP-based sysems [8, 14, 15, 16]. Despe her promse, POMDPs o dae have been lmed o arfcally small oy problems. Sysems n he leraure handle fewer han 10 dsnc user goals [8, 14, 15, 16]. In hs paper, we presen a mehod for scalng up POMDPs for dalog managemen he Summary POMDP mehod and demonsrae s ably o creae a robus -based dalog manager capable of handlng a realscally szed dalog problem. Ths paper s organzed as follows: Secon 2 brefly revews background on POMDPs. Secon 3 presens he summary POMDP mehod. Secon 4 brefly demonsraes he mehod, Secon 5 provdes an evaluaon, and Secon 6 concludes. 2. OVERVIEW OF POMDPS Formally, a POMDP s defned as a uple {S,, T, R, O, Z}, where S s a se of saes; s a se of acons ha an agen may ake; T defnes a ranson probably P ( s s, a ; R defnes he expeced (mmedae, real-valued reward r ( s, a ; O s a se of observaons; and Z defnes an observaon probably, P ( o s, a. The POMDP operaes as follows. each me-sep, he machne s n some unobserved sae s. Snce s s no known exacly, we manan a dsrbuon over saes called a belef sae, b. We wre b(s o ndcae he probably of beng n a parcular sae s. Based on b(s, he machne selecs an acon a, receves a reward r, and ransons o (unobserved sae s, where s depends only on s and a. The machne hen receves an observaon o whch s dependen on s and a. each mesep, we updae b as follows: b ( s = k P( o s, a P( s a, s b( s, (1 s S where k s a normalzaon consan [2]. We refer o mananng he value of b a each me-sep as belef monorng. The varables a, o, and s may be facored no componens. For example, s could be decomposed no s = ( s1, s2, K, sn such ha s1 S1, s2 S2, K, s N S N. In hs case, b(s s wren b( s = b( s1, s2, K, sn, P ( o s, a s wren P( o s1', s2', K, sn ', a, ec. Ths facorng may allow condonal ndependence o be exploed, reducng he parameers requred o express T, R, and Z. The cumulave, nfne-horzon, dscouned reward s called he reurn: = 0 s S λ b ( s r( s, a (2 where b ndcaes he dsrbuon over all saes a me ; b (s ndcaes he probably of beng n sae s a mesep ; a s he acon a aken a me ; and λ s a geomerc dscoun facor, 0 λ 1. polcy π ( b specfes an acon o ake gven a belef sae. 1 The goal of he planner s o fnd a polcy ha maxmzes he reurn. The doman b of a polcy π (b s a pon n S-dmensonal space, called belef space. In general a polcy s a paronng of belef space, where each paron corresponds o an acon. In fac, he sze of he polcy space grows exponenally wh he sze of he observaon se and doubly exponenally wh he 1 We wll assume he plannng horzon for a polcy s nfne unless oherwse saed.

2 dsance (n me-seps from he horzon [2]. Neverheless, realworld problems ofen possess small polces of hgh qualy. In hs work, POMDP opmzaon s performed wh Perseus [11]. Perseus heurscally selecs a small se of represenave belef pons, and hen eravely apples value updaes o jus hose pons nsead of all of belef space, achevng a sgnfcan speed-up over exac mehod. Increasng he number of belef pons enables he algorhm more complex polces a he expense of more compuaon. 3. THE SUMMRY POMDP METHOD Ths paper s concerned wh so-called -fllng dalogs, whch are common n he spoken dalog sysem leraure. In -fllng dalogs, here exs Ν s, where akes on one of N values. The user eners he dalog wh a goal.e., a desred value for each and he am of he machne s o correcly subm he user s goal. Tradonal POMDP-based dalog managers scale poorly because her sae space, acon se, and observaon se grow as Π N. For example, consder a problem wh wo s n whch each akes on one of N 1 = N 2 = M = 1000 values. In hs case, he POMDP s performng plannng on a dsrbuon over M = 10 saes wh more han 10 acons and observaons. Even wh recen echnques, creang polces on POMDPs of hs sze s nracable. By conras, he summary POMDP mehod consders only he mos lkely hypohess for each, effecvely reducng he sze of each o 2 values. The acon and observaon ses are smlarly reduced. Overall hs approach reduces he growh Ν Ν facor from M o 2. The summary POMDP mehod consss of hree phases: consrucon, samplng & opmzaon, and execuon. s he mehod s explaned, an example dalog problem s presened for llusraon Consrucon of he maser POMDP Frs, a POMDP called he maser POMDP s consruced. The sae varable of he maser POMDP s facored no hree groups of varables whch express he user s goal, he user s acon, and he sae of he dalog from he perspecve of he user. Noe ha, from he machne s perspecve, all of hese varables are hdden. Ths facorng faclaes esmang he ranson and observaon funcons and specfyng he reward funcon [14]. The mehod requres ha he sae space componen for he user s goal s furher subdvded no Ν s, wh one componen for each, S1 KS N. The componens for he user s acon and dalog sae may, f desred, be furher decomposed. n llusrave maser POMDP n he ravel doman s shown as an nfluence dagram n Fgure 1, wh componens descrbed n Table 1. In hs dalog problem, he user s ryng o ravel from a from locaon o a o locaon n a world wh 1000 places. Thus he user s goal ncludes 2 componens represenng 2 s, S from and, each of whch akes on 1 of 1000 values. The model of user s acon has been decomposed no he eq 2 componens S, S, S, S, S, and S. Two condonal probably ables form he core of he user model. The dsrbuon P ( s s gves he probably ha he user provdes varous wh-responses (.e., ways of sang a value such as london or o(london for a gven sysem acon. The eq dsrbuon P ( s s, s gves he probably ha he user ncludes a yn-response (yes or no n her uerance for a gven eq sysem acon. The oher user acon componens ( S, S, 2 S, and S are deermnsc funcons and smply enable S and S o be expressed succncly. Fnally, he node S expresses dalog sae,.e., wheher s, from he sandpon of he user, no-saed, saed, or confrmed. In he summary POMDP mehod, he acon se s also facored, no a = a, a, K, a, such ha a S ( 1 from o and a. represens he se of locuonary s avalable o he sysem. In Fgure 1, he sysem acon s formed a = ( a, a, a. conans 5 values: askfrom, confrm-from, ask-o, confrm-o, subm. Compose sysem acons express nclude, for example, confrm-o(london, subm(boson,london, or ask-from. The observaon se O represens he oupu of he speech recognon and parsng process, and ncludes all possble user uerances a he concep level. In he example dalog problem, he observaon se ncludes all combnaons of user acons (as observed by he parser. For example, one observaon mgh be he parse {yes, place(london, o(place(boson}. In general and as n he example, he observaon se wll be oo large o esmae a condonal probably able. Insead, we creae a funcon f ( o, perr P( o s whch esmaes P ( o s for a gven o and per-concep error rae p err. In he example problem, he error rae p err specfes he lkelhood ha a sngle concep s msrecognzed (.e., subsued or deleed. The reward funcon s specfed by he sysem desgner. In he example dalog problem, he reward funcon encourages he sysem o correcly denfy he user s goal as quckly as possble whle observng conversaonal norms. For acons whch do no end he dalog, per-urn penales are assgned whch seek o reward approprae behavor, as lsed n Table 2. For example, a hgher per-urn penaly s gven for confrmng a ha hasn been saed han for confrmng one ha has. The subm acon, whch s he only acon ha ends he dalog, assgns +50 f he user s goal s correcly denfed and -50 f no. The process of belef monorng nvolves nferrng b(s based on he observaon o as n Eq. 1. Inally, b(s s a fla dsrbuon bu as he dalog proceeds, sharpens around he mos lkely values of s. The provson of an accurae user acon model and observaon funcon grealy faclaes hs process Consrucon of he summary POMDP Whereas radonal mehods would aemp o opmze he maser POMDP drecly (e.g., [14], here we form a second POMDP called he summary POMDP n whch opmzaon wll be conduced. The sae space of he summary POMDP, S, conans a componen S N correspondng o each componen S n

3 o from O eq 2 o R O Tmesep Tmesep +1 eq S 2 o o from Fgure 1: Influence dagram showng he maser POMDP for he sample dalog problem, followng [1]. Unshaded nodes are observable, and shaded nodes are unobservable. Crcles are chance nodes; squares are decson nodes; and damonds are uly nodes. rrows show causal nfluence. For clary, only he o s shown, and he acon and uly nodes for he +1 me-sep have been omed. The doed boxes show compose sae and acon varables. Se Meanng S User s goal for eq S Indcaes wheher equals S S Sysem s acon as relaes o hs S -poron of user s acon, e.g., o(x or x S -poron of user s acon (yes, no S acon wh value only, e.g., london 2 S ac w/ name & value, e.g., o(london S Dalog sae: {no saed, saed, or confrmed} O Full concep srng (boh s Sys c: ask-o/from, confrm-o/from, or subm Conen of sysem acon (same se as R Reward (see Table 2 and ex Table 1: Defnon of node labels n Fgure 1. S Dalog sae Sysem acon Reward No saed ask 1 No saed confrm 3 Saed ask 2 Saed confrm 1 Confrmed ask 3 Confrmed confrm 2 Table 2: Per-urn rewards for he sample problem. he maser POMDP. Each s akes on jus 2 values, s { bes, res}, calculaed as follows: s bes, = res, f arg max( b ( s f arg max( b ( s = S S ( (. Essenally, for a gven, he sae of he summary POMDP expresses wheher he mos lkely hypohess n he maser POMDP s correc. Oher sae space componens from he maser POMDP may be ncluded n he summary POMDP sae space f desred. In he example dalog problem, he sae of he summary POMDP S ncludes a componen S whch s a copy of he dalog sae n he maser POMDP, S. The example summary POMDP ncludes a oal of 36 saes. Fgure 2 shows an nfluence dagram of he example summary POMDP, llusrang how a summary POMDP conans fewer and more compac componens han he maser POMDP, faclang opmzaon. The acon space of he summary POMDP ncludes only. Thus, n he example dalog problem, he summary acon ncludes he componen bu no from or o, yeldng a oal of fve acons n he example summary POMDP. The observaon se of he summary POMDP, O, ncludes S O wo componens O S S and O for each. The frs observaon componen s calculaed as follows: c, f o s conssen wh argmax ( b( s S O O = c, f o s nconssen wh argmax( b( s n, f o provdes no nformaon abou b( s Ths conssency relaonshp s specfed by he sysem desgner. In he example dalog problem, f he observaon conans london (whou an ndcaon of o or from, s conssen wh arg max( b ( s = london and nconssen wh any oher value for boh = o and = from. If he observaon conans o(london, s conssen wh arg max( b ( = london and provdes no nformaon abou S from S from S O S S S o S o S O Oo S S Oo R S from S o S from S o S O S S O Oo S S S Tmesep Tmesep +1 Oo s o s from. The R Fgure 2: Influence dagram showng he summary POMDP for he sample dalog problem

4 observaons yes and no alone provde no nformaon abou any S O va O.e., O S O = n. The second observaon componen s calculaed as follows: SS eq, f arg max( b 1( s = arg max( b ( s O = ne, f arg max( b 1( s arg max( b ( s S S In oher words, O ndcaes wheher he mos lkely hypohess for s has changed. For example, n he sample problem, suppose ha arg max( b ( = london, he acon s from confrm-from(london s aken, and no s observed. Ths would resul n a sgnfcan reducon n b + 1( london, and a new mos lkely hypohess, for example arg max( b + 1( s from = leeds. In hs example, london leeds, so O S from S = ne Samplng & Opmzaon To esmae he sysem dynamcs of he summary POMDP, we sample from he maser POMDP usng a random polcy. each mesep, an acon a n he summary POMDP s randomly seleced. The acon a n he maser POMDP s hen formed by seng a = arg max( b( s and combnng a a and, o form a. Fnally, he value of he reward r s calculaed. Nex, a value for he nex (unobserved sae s and observaon o of he maser POMDP are hen sampled. Then he belef sae b (s of he maser POMDP s calculaed as n Eq. 1. Fnally, he sae s and observaons o of he summary POMDP are calculaed as descrbed n 3.2. By samplng repeaedly n hs way, he ranson funcon P ( s s, a, observaon funcon P ( o s, a, and reward funcon r ( s, a of he summary POMDP are esmaed. fer samplng, opmzaon s performed on he summary POMDP o compue a polcy, π ( b ( s a, usng for example [11] Execuon To execue he conroller, belef monorng s performed n he maser POMDP. The belef sae of he summary POMDP s calculaed as s b ( s = bes = max( b( s (3 and ( 1 ( b s = res = b s = bes. The acon a s seleced based on he polcy calculaed above,.e., a = π ( b ( s. The acon a n he maser POMDP s hen formed by seng a = arg max( b( s for all s and combnng a and a, o form a. fer a s aken, a reward r and an observaon o are receved, and he nex belef sae b (s s calculaed (Eq DEMONSTRTION Fgure 5 shows an example conversaon beween he es user model and he POMDP. The small graphs show he dsrbuon of belef mass n b ( s and b ( s over he course of he dalog. Only 3 of he 1000 values of s are shown. he begnnng of he dalog, he belef mass n b ( s s spread evenly over all goals, and ( b s = bes s low. s he dalog progresses, probably mass sharpens around observed values. s explaned n he descrpon of he mehod above, a each me-sep, b ( s = bes = max( b( s. For example, afer urn U1, arg max( b( so = leeds and b ( s = bes = b( leeds. ( b s = res s equal o b for all oher values of s. One srengh of he probablsc approach s ha he sysem never comms o one value for a ; raher, b ( bes s acng as a global confdence score over he course of he dalog. For example, he no observed n U4 redsrbues probably mass n ( b s o from s o = leeds o all oher values, causng a decrease n b ( so = bes. lernavely, a second observaon provdng furher suppor for s from = cambrdge n U5 causes an ncrease n b ( s from = bes. second srengh s ha he effec on b of an observaon s scaled by he lkelhood of he correspondng user acon. For example, n U4, he user acon from(sheffeld s very unlkely, so when he observaon from(sheffeld s made ncreases b( s from = sheffeld bu only slghly. 5. EVLUTION We are neresed n deermnng how performance of he summary mehod vares wh error rae, how robus he mehod s o varaons n user behavor, and fnally how he summary mehod compares o radonal mehods. To do hs, we employ real dalog daa from he SCTI-1 corpus [13]. The SCTI-1 corpus conans 144 human-human dalogs n he ravel/ours nformaon doman usng a smulaed SR channel [12]. The corpus conans a varey of word error raes, and he behavors observed of he subjecs n he corpus are broadly conssen wh behavors observed of a user and a compuer usng a real speech recognon sysem [13]. The corpus was segmened no a ranng sub-corpus and a es sub-corpus, whch are each composed of an equal number of dalogs, he same mx of word error raes, and dsjon subjec ses. Wzard/User urn pars were annoaed, and one user model was hen esmaed from each sub-corpus. The ranng user model was nsalled n nodes S and S n he maser POMDP, gvng he wh poron of he user s response such as o(cambrdge, and he yn (yes/no poron, respecvely. To creae he summary POMDP, 20,000 dalog urns were sampled. Polcy opmzaon was performed wh 50 belef pons, 50 eraons, and a dscoun of λ = To form a baselne, 20,000 dalog urns were hen run wh he 2 The number of belef pons broadly ses an upper bound on he complexy of he resulng polcy.

5 Reward ganed / urn verage or expeced reurn User model bul from ranng se User model bul from es se Fgure 3: p err vs. reward ganed per urn for he sample dalog problem p err M (Number of dsnc values Summary POMDP Baselne Fgure 4: n (number of dsnc values vs. average or expeced reurn for a smplfed 1- dalog problem. The baselne s a drec soluon of he maser POMDP. resulng polcy and he ranng user model. Nex, he es user model was nsalled no he Maser POMDP, and 20,000 dalog urns were run wh he polcy creaed from he ranng user model. Ths process was repeaed for values of p err from 0.05 o Fgure 3 shows resuls for a range of values of p err. The Y- axs shows average reward ganed per dalog urn, and error bars ndcae he 95% confdence nerval for he rue average reward ganed per dalog urn. s speech recognon errors ncrease, he average reward per urn decreases, conssen wh radonal mehods [8, 14, 15]. For all values of p err excep 0.05, performance on he es user model s no less han 0.5 pons of he ranng user model and s generally very close. In oher words, he polcy creaed for he ranng user model s performng smlarly on he esng user model, mplyng ha he mehod s reasonably robus o varaons n paerns of user behavor. In some cases, e.g., p err = 0.05, he polcy performs beer on he es user model han on he ranng user model. Ths s possble because dfferen user models can provde dfferen amouns of nformaon. In hs example, he es user model provdes slghly more nformaon han he ranng user model, whch enables he polcy o perform beer on he esng user model a ceran error raes. Fnally, he sysem was run n a smpler one confguraon whch allowed he maser POMDP o be opmzed drecly. Drec soluons were compued wh 1,000 belef pons and 50 eraons. The summary POMDP mehod was hen appled usng 20,000 dalog urns, 50 belef pons and 50 eraons. Ths process was repeaed for values of M (.e., dsnc number of values from 3 o The concep error rae was se o p err = 0.30, and he dscoun o λ = for all expermens. Fgure 4 shows M vs. average or expeced reurn for he summary mehod and he baselne. The soluon algorhm was no able o opmze he maser POMDP drecly for M > 100. Error bars show 95% confdence nervals for rue expeced reurn. For small problems,.e., lower values of M, he summary mehod performs equvalenly o he baselne. For larger problems, he summary mehod ouperforms he baselne by an ncreasng margn. Moreover, he summary POMDP polcy was derved usng 95% fewer belef pons han he baselne;.e., he summary mehod s polces scale o large problems and are much more compac. I s neresng o noe ha as M ncreases, he performance of he summary POMDP mehod appears o ncrease oward an asympoe. Ths rend s due o he fac ha all confusons are equally lkely n hs model. For a gven error rae, he more conceps n he model, he less lkely conssen confusons are. Thus, havng more conceps helps he polcy denfy spurous evdence over he course of a dalog. 6. CONCLUSION POMDP-based dalog managers have been shown o ouperform MDP-based dalog managers, bu o dae have been lmed o arfcally small problems. Ths paper has demonsraed a mehod for scalng POMDP-based dalog managers o problems of a realsc sze. The mehod has been demonsraed wh a 2- problem ncorporang a user model esmaed from real dalog daa, and expermens have shown ha he resulng dalog manager copes well wh changes n paerns of user behavor. Moreover, summary dalog polces appear o be as good as hose derved drecly for he full model. s presened here, he sze of he sae space and observaon space n he summary POMDP sll grow exponenally n he number of s. Whle many real-world problems use a handful of s (e.g. 2, 3 or 4, ohers use many s. Thus one heorecal ssue o address n fuure work s how o scale o large numbers of s. The facored naure of he summary POMDP may be of some help here, for example [7]. Fnally, fuure praccal work wll aemp o valdae he mehod by consrucng an end-o-end sysem. 7. CKNOWLEDGEMENTS The auhors would lke o hank Pascal Poupar for several helpful, nsghful dscussons. Thanks also o Ma Sule and Jos Schazmann for helpful commens on he presenaon of hs paper. The work repored n hs paper was suppored by he EU FP6 Talk Projec. 11. REFERENCES [1] Fnn V. Jensen. Bayesan Neworks and Decson Graphs. New York: Sprnger Verlang, 2001.

6 [2] Lesle Pack Kaelblng, Mchael L. Lman and nhony R. Cassandra. Plannng and cng n Parally Observable Sochasc Domans. rfcal Inellgence, Vol. 101, [3] Saffan Larsson and Davd Traum. Informaon sae and dalogue managemen n he rnd dalogue move engne oolk. Naural Language Engneerng, 5(3 4: , [4] Esher Levn, Robero Peraccn, and Weland Ecker. Sochasc Model of Human-Machne Ineracon for Learnng Dalogue Sraeges. IEEE Transacons on Speech and udo Processng, Volume 8, No. 1, 11-23, [5] Esher Levn and Robero Peraccn. Sochasc Model of Compuer-Human Ineracon For Learnng Dalogue Sraeges. Eurospeech, Rhodes, Greece, [6] Olver Pequn. Framework for Unsupervsed Learnng of Dalogue Sraeges. Ph D hess, Faculy of Engneerng, Mons, Belgum, [7] Pascal Poupar and Crag Bouler. VDCBPI: an pproxmae Scalable lgorhm for Large Scale POMDPs. dvances n Neural Informaon Processng Sysems 17 (NIPS- 2004, Vancouver, BC, pp Walker. Opmzng Dalogue Managemen wh Renmen Leanng: Expermens wh he NJFun Sysem. Journal of rfcal Inellgence, Vol. 16, , [11] Mahjs T. J. Spaan and Nkos Vlasss. Perseus: randomzed pon-based value eraon for POMDPs. Techncal Repor IS-UV-04-02, Informacs Insue, Unversy of mserdam, [12] Mahew Sule, Jason D. Wllams, and Seve Young. Framework for Wzard-of-Oz Expermens wh a Smulaed SR-Channel. Inernaonal Conferences on Spoken Language Processng (ICSLP-2004, Jeju, Souh Korea, [13] Jason D. Wllams and Seve Young. Characerzng Task- Orened Dalog usng a Smulaed SR Channel. Inernaonal Conference on Spoken Language Processng (ICSLP, Ocober 2004, Jeju, Souh Korea. [14] Jason D. Wllams, Pascal Poupar, and Seve Young. Facored Parally Observable Markov Decson Processes for Dalogue Managemen. 4h Workshop on Knowledge and Reasonng n Praccal Dalog Sysems, Inernaonal Jon Conference on rfcal Inellgence (IJCI, ugus 2005, Ednburgh. [8] Ncholas Roy, Joelle Pneau and Sebasan Thrun. Spoken Dalogue Managemen Usng Probablsc Reasonng. nnual meeng of he ssocaon for Compuaonal Lnguscs (CL [9] Konrad Scheffler and Seve Young. uomac learnng of dalogue sraegy usng dalogue smulaon and renmen learnng. Proc. Human Language Technology (HLT-2002, San Dego, pp [10] Sander Sngh, Dane Lman, Mchael Kearns and Marlyn [15] Jason D. Wllams, Pascal Poupar, and Seve Young. Parally Observable Markov Decson Processes wh Connuous Observaons for Dalogue Managemen. In Proc. 6h SgDal Workshop on Dscourse and Dalogue, Sepember 2005, Lsbon. [16] Zhang Bo, Ca Qngsheng, Mao Janfeng, and Guo Banng. Plannng and cng under Uncerany: New Model for Spoken Dalogue Sysem. Proceedngs of he 17h nnual Conference on Uncerany n rfcal Inellgence (UI-01. San Francsco, US, s from s o Turn Noes b ( s from b ( b ( s o b ( [pror o dalog sar] S1: Where are you gong o? U1: London [msrecognzed as Leeds] S2: Where are you leavng from? U2: From Cambrdge [msrec as To Oxford] S3: Where are you leavng from? U3: From Cambrdge [reco ok] S4: To Leeds, s ha rgh? U4: No, o London [msrec as No, from Sheffeld] S5: Where are you gong o? U5: To London from Cambrdge [reco ok] [Sysem prns cke from London o Cambrdge] Pror o dalog sar, probably mass s spread evenly over all values. Top hypohess for o s now leeds; from has receved no nformaon. User model predcs hs obs. s unlkely: o(oxford ges mnmal belef mass. Top hypohess for from s now cambrdge. o(leeds reduced; from(sheffeld has small effec b/c of user model. Top hypohess for o s now london; from s now very ceran. CMB. CMB. CMB. CMB. CMB. CMB. Fgure 5: Example conversaon and belef saes. The user s ryng o ravel from Cambrdge o London.