XMAP: Track-to-Track Association with Metric, Feature, and Target-type Data

Transcription

1 XMAP: Track-o-Track Aocaon wh Merc, Feaure, Targe-ype Daa J. Ferry Meron, Inc. Reon, VA, U.S.A. Abrac - The Exended Maxmum A Poeror Probably XMAP mehod for rack-o-rack aocaon baed on a formal, Bayean mehodology for ncorporang merc, feaure, arge-ype daa. The merc componen mprove upon he clacal dervaon of he adapve hrehold o produce a more robu alernave, whch can hle cluer wh very few rack rack wh large covarance. The feaure arge-ype componen are reaed only, allowng for he pobly ha he performance of he feaure exracor depend on arge ype. Th couplng allow feaure nformaon o be nerpreed dfferenly dependng on he reul of a arge clafer from a feaure meauremen beng deemed accurae whn whn a mall olerance, o he meauremen beng hrown ou alogeher. A key nnovaon n he dervaon he non-nformave noe aumpon ued n he feaure meauremen model, whch gve a mple, robu form o he reul. Keyword: Daa aocaon, feaure, arge ype, adapve hrehold, noe model. Inroducon The XMAP exended Maxmum A Poeror probably mehod [] a prncpled, Bayean echnque for compung he opmal rack-o-rack aocaon of daa. I begn wh mple, general aumpon abou he naure of he probably drbuon of a a e of arge abou whch one aempng o gan nformaon, b he error made n gaherng h nformaon. We wll aume here are a number of enor ofen, bu no necearly, phycal enor, ha on each enor here are a number of rack, each of whch provde nformaon abou ome arge. The XMAP mehod produce a formula for he probably of each poble aocaon of he rack on he varou enor. An aocaon a pecfcaon of whch rack refer o a common arge. Geng he correc aocaon a neceary prereque o fung he daa from he enor: ncorrec aocaon lead o he fuon of mmached nformaon. The aocaon eleced by he XMAP mehod may uly be called opmal: armed wh he formula for he probably of each aocaon, he XMAP mehod chooe he aocaon wh maxmal probably. XMAP an exenon of Mor Chong MAP procedure []. The MAP mehod wa formulaed wh only merc daa n mnd, n he wo-enor cae recover he radonal oluon o he aocaon problem [3] nvolvng a co marx whoe enre are χ dance excep ha produce naurally a hrehold for aocaon. Oher mehod eher force a many aocaon a poble, or have ome ad hoc cheme for deermnng wheher he co of aocang wo rack uffcenly low o aocae hem. The formulaon of XMAP may be dvded no wo par. The fr he dervaon of he probably of an aocaon, whch preened n Appendx A. Th dervaon eenally equvalen o he dervaon of MAP gven n [4], bu mpler a employ le mahemacal machnery uch a rom e. The chef dfference n he dervaon of XMAP ha work wh an abrac pace raher han a merc pace. The econd par of he dervaon of XMAP gvng a ueful rucure o he abrac pace employ, reaonable aumpon abou he probable on ha pace. Th laer par he opc of h paper. We make he followng aumpon abou he daa. We aume ha he arge are ndependenly dencally drbued..d. ha here are no pl or merged rack:.e., each arge produce a mo one rack on a gven enor, each rack on a gven enor are from a mo one arge. We aume here are no fale alarm. We aume ha he error of each rack depend only on he enor /or nformaon carred by he rack uch a a covarance marx for a Gauan drbuon. See Bar-Shalom Chen [5] for he exenon o rack wh correlaed error. Fnally, we aume ha he pror on he number of arge Poon drbued wh mean ν. See Mor Chong [4] or Appendx A for he exenon o non-poon drbuon. Under hee condon, he probably of any aocaon proporonal o he produc of facor R,, where he produc run over all par, uch ha he aocaon par rack on enor wh rack on enor. For,, he meauremen receved on rack of enor. We ue meauremen n a broad ene here, encompang any knd of daa en o he aocaon algorhm n parcular meauremen could refer o he mean of a poeror

2 drbuon obaned from he fuon of prevou rack nformaon wh a enor meauremen updae. Each facor R, gven by he formula R, P,, P νp where he probable P P, are defned o be P P, P DxQ 3 D xl x p 0 x dx, P DxP DxL x L x p 0 x dx. 3 Thee equaon are derved n Appendx A, where hey appear n lghly dfferen noaon a equaon In 3 PD x he probably of enor deecng a arge n ae x, wherea Q D x P D x he probably of non-deecon. The meauremen lkelhood funcon L x he probably deny for a deeced arge n ae x producng he meauremen on enor. Fnally, he pror probably deny of a arge beng n ae x denoed p 0 x. Hence P, he probably ha rack on enor on enor are from he ame arge, wherea P he probably ha rack on enor no deeced on he oher enor enor 3. Therefore R, he probably rao beween he hypohee ha rack are from he ame veru dfferen arge, he facor of ν beng due o he hypohe of dfferen arge requrng one exra arge n oal. The MAP aocaon he one wh maxmal probably. Fndng h aocaon a ard neger programmng problem. I may be olved wh he Jonker Volgenan Caañon JVC algorhm [6], for example. The key o he XMAP procedure n defnng he ae meauremen varable x, her probably rucure. We begn h defnon by expreng he ae a x x M, x J, where x M he merc ae x J he on arge-ype/feaure ae. Smlarly, he meauremen wll be aumed o con of merc on arge-ype/feaure componen: M, J. Aumng ha he merc arge-ype/feaure pror meauremen lkelhood funcon are ndependen, hen we may decompoe he probably rao R, a R, ν R M M, M R J J M, J. 4,.e., Secon develop a formula for R M he merc componen of XMAP. The reul of h econ a refnemen of he adapve hrehold of he MAP procedure, whch acheved by reang more carefully one of he ard aumpon ha made n he MAP dervaon., M, J,.e., Secon 3 develop a formula for R J J he on arge-ype/feaure componen of XMAP. IN h econ we conder a very general cenaro, n whch a clafer employed o aceran arge ype, he feaure meauremen lkelhood funcon allowed o depend rongly on h arge ype. Alhough XMAP work well when combnng feaure clafer nformaon under he aumpon ha hey may be reaed ndependenly no only from he merc bu from each oher a well, ofen he cae ha he clafcaon provde mporan nformaon abou he drbuon of he underlyng feaure ae for a gven feaure meauremen. Secon 3 preen he mehodology ha allow h nformaon o be exploed n a mahemacally correc manner. Merc componen We wre he merc componen n 4 a R M R, P,, 5 P where, droppng wha would be a ubquou upercrp M, P L x p 0 x dx, 6 P, L x L x p 0 x dx. 7 Here we have aumed ha he deecon probably ndependen of he merc ae: wll be ncorporaed no R J n Secon 3. There are wo ard aumpon made abou merc nformaon. Fr, he meauremen lkelhood funcon are aumed o be Gauan. Accordngly, we le L x N ; x, V,.e., a Gauan wh mean x covarance marx V. Second, he pror drbuon on he ae aumed o be p 0 x I S x/ VolS, where S ome regon of pace n whch he arge are aumed o le, I S he ndcaor funcon for he regon.e., I S x for x S 0 oherwe. Wh hee aumpon we ge R, VolSN ;, V + V. 8 The ard way of producng he pror volume VolS o emae from he daa. Beyond he heorecal problem h poe n volang Bayean mehodology, here he praccal problem ha he preence of ouler can wreak havoc wh a volume emaor. A more flexble model ha he rack are cluered no everal regon n whch he unform pror aumpon vald. Therefore one fr mu cluer rack, hen apply XMAP o each cluer ndvdually. Th nroduce he requremen ha he volume emaor funcon robuly even on a cluer of very few rack. Th concern addreed n Secon... Volume emaor To conruc a repreenave pror regon S we fr relabel all meauremen from boh enor { } N. P

3 The cener of he regon S we ake o be ˆx N N. 9 If he merc daa con of poon only, hen a enble hape o precrbe for S an ellpod. To ge he hape of h ellpod we compue he covarance marx V of he meauremen { }: V kl N N k ˆx k l ˆx l. 0 One can alo compue he covarance marx of he ellpod S by replacng he um over all { } n 0 wh an negral over all pon n S. We chooe S o be he unque ellpod wh covarance marx V cenrod ˆx. The volume of S VolS πθ n/ de V, where n he number of dmenon of he merc daa, θ + m/ m/! /m. We reerve he opon of choong m n o be a dvor of n raher han n elf, leng S be he Carean produc of m-dmenonal ellpod. E.g., we could have m 3 n 6 f he merc daa con of 3-d poon velocy. In general, we e m o be he number of dmenon of he underlyng phycal pace. The value of θ for m,, 3 are 6/π.9,, 5/3 3 6/π.07, repecvely. Ung h volume emae we may rewre 8 a R, θ n/ de V de V + V exp T V + V 3. There are ome problem wh 3, however. When N n, de V 0, o R 0, precludng he aocaon of any rack a all. Furhermore, when V become large, we alo have R 0, precludng he aocaon of any rack wh large varance. A oluon o he fr problem mgh be o modfy he volume emae omehow o ha alway produce pove value, a oluon o he econd mgh be o conran he covarance marce V no o exend beyond S. Bu nead of cobblng uch ad hoc fxe ono 3, we addre he roo caue of hee problem.. Ellpodal mehod The ard mplfcaon made n dervng 8 replacng he negral over S wh an negral over all pace. Th mplfcaon clearly napproprae when he varance V of a rack exend beyond he regon S. Inegrang exacly over S no an appealng alernave. However, we can approxmae he negral over S n a manner ha repec fne exen whle producng a mple reul. The appeal of approxmang p 0 x by / VolS ha h a homogeneou funcon: nvaran under all paal ranlaon. Only he conan funcon ha h propery, o ung any oher funconal form for p 0 x wll drup he homogeneou form of he reul: e.g., R mgh depend no only on, bu on he dance of from ˆx a well. Therefore we approxmae p 0 x no by a ngle funcon of x, bu by a kernel fx c, where he hf c ranlae he kernel o wegh he domnan regon of he negr. Th eay o do n 6 7 becaue he negr proporonal o a Gauan n each cae he produc of wo Gauan beng proporonal o a Gauan n he cae of 7, o c choen o be he mean of h Gauan. For f o have a hape mlar o ha of p 0, he varance of f hould be proporonal o ha of p 0,.e., o V. In order o ge mple reul, we chooe f o be proporonal o a Gauan. Thu we model f a fx c CN x; c, k V. Two pulaon we place on f ha yeld he correc reul n he lm V V V V. The requremen mpled by he fr lm ha C ; he requremen mpled by he econd ha f0 p 0 0 / VolS, whch n urn mple k θ. The reulng formula for R R, de I + M de I + M de I + M + M 4 exp T V + V, where M θ V V. Noe ha 4 doe no degenerae for large value of V or ngular marce V. Even when a cluer con of only one or wo rack, 4 gve enble reul, n conra o 3. Co Track Error Fgure : Comparon of approxmaon Conan pror Gauan pror Exac The accuracy of 4 lluraed by Fgure, whch compare he co for a he radonal approxmaon of he pror by a conan funcon b he Gauan approxmaon o he co for c he exac reul for -d daa. In h cae, VolS 0.6, he co ploed agan he ard devaon of he rack error. The Gauan approxmaon accurae acro he enre range of rack error, wherea he radon mehod fal when he rack error are oo bg relave he gven value of VolS. The effecvene of h mehod of compung a hrehold depend on how reaonable a f he un-

4 form drbuon over ome regon o he daa. Becaue uch a f can be farly poor, a beer f ofen produced by cluerng he daa, hen agnng a unform pror over each cluer. In x-dmenonal poon velocy pace, h requre a lea even daa pon o deermne a pove volume. However, he ellpodal mehod decrbed here requre only wo pon n a cluer. Thu compleely general becaue a one-pon cluer preumably already reolved a no beng aocaed. The performance of uch a mehod would depend on he cluerng algorhm employed. I would be ueful o compare he performance of h mehod for varou cluerng algorhm wh alernave approache uch a n u unng cheme. 3 Type/feaure componen In addon o merc nformaon, wo oher knd of nformaon ofen avalable for daa aocaon are arge-ype feaure. There a fear aocaed wh ncorporang uch daa no a merc-only aocaon. Merc aocaon a well eablhed echnology, he mproper ncorporaon of arge-ype or feaure daa can lead o erou performance degradaon. However, when hled correcly, addonal nformaon only mprove performance. There are hree key o he ncorporang h daa properly. Fr, one mu have an approprae model of he meauremen lkelhood funcon for he arge-ype /or feaure. Second, one mu proce ha lkelhood model correcly no a co. Thrd, one mu accoun for he pobly of he daa no fng he model. Meeng he fr requremen requre acce o large expermenal or mulaed daa e for he feaure of nere. The mehodology for exracng a lkelhood model from uch daa no addreed here. Wha addreed are he econd requremen, whch he opc of he remander of h econ, he hrd, whch addreed n Secon 3.. A arge ype a dcree varable repreenng a fne number of clae o whch he arge could belong. I ypcally exraced.e., meaured by a clafer. The poble clafcaon call c are uually he ame a he poble arge ype, bu for he purpoe of daa aocaon here no requremen ha he wo are relaed. The qualy of a clafer deermned by confuon marx. We ue L c o denoe he confuon marx enre for enor. A feaure ypcally a ngle real-valued varable or perhap a real array whch meaured by a feaure exracor. When he feaure ndependen of arge ype, he exracor characered by he meauremen lkelhood funcon L w y: he probably deny of exracng he value w when he rue feaure ae y. Here, however, we aume ha he feaure arge ype varable are nerrelaed. We denoe he on arge-ype/feaure ae meauremen varable x J x, y J c, w, repecvely. The on arge-ype/feaure componen n 4 may be wren R J R, P, P, P 5 where P P DxQ 3 D xl x p 0 x dx, 6 P, PDxP Dx L x L x p 0 x dx. 7 The meauremen lkelhood funcon do no explcly depend on he rack ndex here unlke he merc cae becaue here no addonal nformaon beng ored wh he rack uch a a covarance marx V needed by he meauremen lkelhood funcon. Gven he repreenaon x, y c, w, we may expre he requred pror meauremen lkelhood funcon a p 0, y p 0 p 0 y, 8 L c, w, y L c, y L w c,, y, 9 where c w are he called arge ype exraced feaure value, repecvely, for rack on enor. We aume ha hee, along wh he merc daa, are alway agned o he correc rack. There are no mplfcaon made n 8 9: hey are u manpulaon of condonal probably. We mplfy 9 by makng hree aumpon. Fr, we aume ha L c, y ndependen of y. Alhough he clafer behavor may n fac depend on he feaure ae y, we aume ha h a mnor effec. Th convenen becaue alhough one expec a confuon marx L c o be provded wh a clafer for a gven enor confguraon, one unlkely o be provded wh he dependence of he clafer on y. Second, we aume ha L w c,, y ndependen of c. In oher word, we aume ha he rue arge ype uffce o deermne how he meauremen w depend on he feaure ae y, ha he called ype yeld lle addonal nformaon. Fnally, we aume ha he behavor of he feaure exracor no calbraed o he enor confguraon. Th aumpon no really neceary o he dervaon, farly eay o allow for uch a dependence. However, h preenaon of XMAP a amed a praccal applcaon, deemed unlkely ha he proce of modelng he feaure exracor would be performed o horoughly a o ncorporae enor confguraon. Wh hee aumpon, 9 mplfe o L c, w, y L c L w, y. 0 Fnally, we aume ha he deecon probably depend only on he arge ype: P D, y P D.

5 Ung 8, 0, we may mplfy 6 7 o P c, w PDQ 3 D L c p 0 P w, where P c, w, c, w PDP D L c L c p 0 P w, w, 3 P w L w, y p 0 y dy, 4 P w, w L w, y L w, y p 0 y dy. 5 I reman o evaluae he negral n 4 5. The fr ep n dong o nvolve makng a very general aumpon abou he form of he meauremen noe whch we call he non-nformave noe aumpon. Th ep enure he robune of he procedure whle producng very mple formula. 3. Non-nformave noe A Gauan a ypcal model for Lw, y, he probably of a arge of ype n feaure ae y generang feaure meauremen w. Such a model carre he rk of reurnng ncredbly ny aemen of probably for a meauremen w arng e.g., 0 00, or maller when doen mach y well. In a realc uaon, he probably of a meauremen w could never be ha mall becaue here alway he pobly of ome glch n he feaure exracon roune. By allowng uch ny probable o occur n he model, one run he rk of he feaure exracor compleely prevenng a par of rack beng aocaed even when he merc nformaon exremely favorable o aocaon. Becaue one of he chef fear n ncorporang a feaure exracor no an aocaon algorhm ha mgh run merc-only performance ha already farly good, que pruden o accoun for he pobly of noe n he feaure meauremen model. Noe: one could make he ame argumen o pon ou ha he rapd decay of Gauan allow anomalou merc daa o overrde perfec feaure mache, o one mgh nclude a noe erm n he merc aocaon erm a well. Th would be a pon worh conderng when feaure exracon echnology reache he maury ha merc rackng enoy. We regard a funcon a repreenng pure noe when he drbuon of meauremen w yeld ndependen of he acual feaure ae y. One opon for a noe model a unform drbuon of w over a ceran range. Alhough h eem mple, nroduce an addonal parameer he wdh of he drbuon complcae he requred negral, bu no necearly a good model of noe. We erm he noe model employed n XMAP he non-nformave model. Th model e he drbuon of meauremen due o noe equal o he overall drbuon of meauremen. I called non-nformave becaue a meauremen w provde no nformaon a o wheher aroe from a clean meauremen of ome arge or merely from noe. Were one o know how noe dffer acally from clean meauremen, one could ue h nformaon o flag ceran meauremen w a more lkely o have aren from noe han oher, perhap queee even more performance ou of an aocaon algorhm, bu a he rk of algorhm robune hould he noe behave dfferenly han expeced. In conra, he non-nformave aumpon provde a conervave, robu baelne model for noe. To ue h model, we expre he meauremen lkelhood funcon a L w, y a µ L µ w, y + a µ P w, 6 where L µ w, y he modeled erm of Lw, y, a µ he wegh gven o h erm. The remanng wegh a µ gven o he non-nformave noe erm P w defned n 4. Seng a µ 0 ndcae ha he feaure doe no ex or canno be exraced for arge ype. We defne he followng analog of 4 5 for L µ : P µ w L µ w, yp 0 y dy, 7 P µ w, w L µ w, y L µ w, y p 0 y dy. 8 Becaue he drbuon of he noe mply P w whch he drbuon he noe non-noe ogeher, he drbuon of he non-noe alo P w P w P µ w, 9 whch can be een by ubung 6 no 4. The value of P w, w a lle more nereng: P w, w a µ P µ w, w + a µ P µ w P µ w. 30 A foruou conequence of he non-nformave noe aumpon ha only lghly complcae he requred compuaon. The quany P w dencal o P µ w, wherea P w, w a mple combnaon of P µ w, w, P µ w, P µ w. To complee he formulaon, we now compue hee P µ quane for a repreenave meauremen lkelhood model. 3. Reul The fnal ep of he dervaon modelng he feaure pror a p 0 y I y, 3

6 where I he ndcaor funcon over an nerval of lengh. We employ he ard mplfcaon of exendng negral over h nerval o he enre real lne, whch vald provded farly large compared o he meauremen error defned by L µ w, y. Th mplfe 7 8 o P µ w, 3 P µ w, w K w, w, 33 where K w, w L µ w, y L µ w, y dy. 34 We now collec he equaon above no a ngle, mple reul. The goal a formula for R J, whch gven by 5. Th may be mplfed o R J R c, w, c, w P c, w, c, w P c P c, 35 where 3 have been mplfed o P c p 0 P DQ 3 D L c, 36 P c, w, c, w p 0 P DP DL c L c a µ + a µ K w, w, 37 repecvely. Depe he mplcy of he equaon 35 37, hey are que powerful, for hey ndcae he proper way o combne he pror drbuon of arge ype p 0, he deecon probable PD for each enor arge ype, he pror pread of he feaure value for each ype, he wegh a µ o agn o he model for each ype, wh he confuon marce L c for each enor, he kernel K w, w ha ncorporae he feaure meauremen model for each arge ype. Th may be more freedom han one dere, n whch cae he pror on arge ype p 0 could be e o a unform drbuon, he deecon probable PD o a ngle value P D, he pror feaure pread o a ngle value. The value L c, on he oher h, are a propery of he clafer, are deermned when he clafer calbraed. Smlarly he value of a µ K w, w are propere of he feaure exracor. We now gve an example of K w, w for a pecfc, hypohecal uaon. 3.3 Example: bmodal feaure model The feaure model hould be deermned from Mone Carlo mulaon or mlar ude of he feaure exracor over a range of realc condon. To gve a concree example, we conder a hypohecal example n whch oberved ha for every arge ype, he meauremen ofen cluer near he rue feaure value, bu omeme cluer abou a conan offe from he rue value. In h hypohecal uaon, we fnd: ha he offe δ he ame for every arge ype; ha he varance abou he rue offe value are roughly he ame a each oher, bu vary wh ; ha he probable of fallng near he rue value, offe value, or elewhere vary wh a well. In uch a uaon, he followng meauremen lkelhood model would be approprae: L µ w, y C 0 N w; y, V +C N w; y +δ, V, 38 where C 0 + C. The emprcal udy would deermne all of he conan nvolved, a well a a µ. E.g., a µ repreen he fracon of meauremen fallng near he rue or he offe value, ec. Aumng V, vald o ue apply he mplfcaon of exendng negral over he enre real lne, whch wa nvoked n dervaon of 35. Ung h bmodal model, we fnd K w, w A 0 N w; 0, V + A N w; δ, V + N w; δ, V, 39 wh w w w, A 0 C 0 + C, A C 0 C. Th complee he explc pecfcaon of he probably rao R J for he on arge-ype/feaure componen of he XMAP mehod. 4 Concluon The XMAP procedure for rack-o-rack aocaon able o ule any knd of daa n a mahemacally rgorou way provded one able o model he pror meauremen lkelhood funcon for he daa. Secon develop he merc componen of XMAP, whch mlar o he ard formulaon, bu ue a dfferen adapve hrehold whch properly reflec he fne varance of he pror drbuon of he merc ae. Secon 3 how how o ncorporae on feaure/arge-ype daa n a manner ha explo he confuon marce ypcally provded for a arge-ype clafer, whle provdng he uer wh he opporuny o oban mproved aocaon reul by formulang meauremen model for he feaure exracor for each arge ype. Typcally, one would ue a ngle model for group of arge ype wh mlar behavor he power of h formulaon come from beng able o ue he clafer o dnguh beween gro dfference n behavor of he feaure meauremen proce. Becaue he probablc rucure of he daa ncorporaed correcly, here no danger of performance degradaon veru merc-only aocaon, provded one provde he correc meauremen lkelhood funcon confuon marce for he daa. In parcular, he non-nformave noe aumpon lend grea robune o he reul produce mple formula a well.

7 Reference [] L. D. Sone, M. L. Wllam, T. M. Tran. Track-o-rack aocaon ba removal. In SPIE AeroSene Inernaonal Conf., Aprl 00. [] S. Mor C.-Y. Chong. Effec of unpared obec enor bae on rack-o-rack aocaon: Problem oluon. In Proc. MSS Senor Daa Fuon, volume, page 37 5, 000. [3] S. Blackman R. Popol. Degn Analy of Modern Trackng Syem. Arech, Boon, 999. [4] S. Mor C.-Y. Chong. Evaluaon of daa aocaon hypohee: Non-poon..d. cae. In 7 h Inernaonal Conf. on Informaon Fuon, 004. [5] Y. Bar-Shalom H. Chen. Mulenor rack-orack aocaon for rack wh dependen error. In 43rd IEEE Conf. on Decon Conrol, 004. [6] R. Jonker A. Volgenan. A hore augmenng pah algorhm for dene pare lnear agnmen problem. Compung, 38:35, 987. A Probably rao dervaon A. Sngle enor In h econ we compue he probably P, a x, n of geng a pecfc array of meauremen value on a enor, geng hem n he way precrbed by he arge-o-rack map a, gven ha here are n arge whoe ae are led n he array x. To begn, we aume here a fne e J {,,..., n} of arge whoe ae are gven by x {x, x,..., x n }. The probably of a enor deecng a arge n ae x denoed PD x, Q D x P D x. For he gven enor, he probably of deecng a arge aumed o depend only on he ae x of he arge. Gven hee aumpon, he probably of he ube of arge deeced by enor beng J P J x, n J P Dx J\J Q Dx. 40 We aume ha each deeced arge produce exacly one rack on he enor no pl or merged rack, ha rack canno are whou arge producng hem no fale alarm. Thu here a oneo-one correpondence beween he e of deeced arge J he rack on enor. Leng n J, we defne I {,,..., n } o be he rack ndce, le a : J I denoe a becon from arge o rack. We aume ha all uch becon are equally lkely,.e., ha P a J, x, n / n!. Mulplyng h expreon by 40 yeld an expreon for P a, J x, n, whch may be wren more mply a P a x, n, nce a deermne J : P a x, n n! J P Dx J\J Q Dx. 4 Now le denoe he meauremen on rack on enor, he array of all meauremen. Smlarly, le L x, denoe he meauremen lkelhood funcon for rack on enor :.e., he probably or probably deny of a deeced arge n ae x producng he meauremen. Gven he n ae x, he map whch agn a ube of hem o he rack on enor, he probably of geng he array of meauremen P a, x, n L a a x. 4 J Mulplyng 4 4 yeld he dered quany: P,a x, n n! J P Dx L a a x A. Mulple enor J\J Q Dx. 43 I raghforward o generale 43 from a ngle enor o a e of enor S. We le S denoe he array of all ndvdual meauremen array, mlarly le a a S denoe all arge-o-rack map. Aumng he meauremen proce ndependen for each enor, we have P,a x, n n! S 44 PDx L a a x Q Dx. J J\J Alhough a defne whch arge produce whch rack, more convenen o expre he nformaon conaned n a dfferenly. We defne a {, : a }, 45 a S { : J }. 46 The funcon a S gve he e of enor whch deec each arge, wherea a gve he rack ndce of each deecon oo. In h new noaon we expre 44 a P,a x, n n! S J 47 PDx L x Q Dx., a S\a S We may elmnae he dependence on x from 47 by negrang over he produc of 47 p 0 x n. We aume ha he ndvdual arge ae are..d. o p 0 x n he produc of p 0 x over J. Hence P, a n p 0 x P, a x, n dx. 48 J Th negral over x eparae no a produc of negral over each x. Thee negral are parcularly

8 mple for arge ha are no deeced by any enor: each uch negral equal o q p 0 x Q Dx dx. 49 S We le J D { J : a } 50 be he e of deeced arge, n D J D be he number deeced. We may now mplfy 48 o P,a n n q n n D p 0 x! S J D 5 PDxL x Q Dx dx., a S\a S A.3 Aocaon probably The funcon a defned n 45 map each arge o he cluer of rack produce. An aocaon [a] defned o be a e of uch cluer, [a] { a : J D }. 5 Dfferen funcon a a can yeld he ame aocaon:.e., [a] [a ]. Indeed, here are exacly n! / n n D! funcon whch yeld he aocaon [a], all of whch are equally probable, o P, [a] n n! n n D! P, a n. 53 If he pror probably for he number of arge beng n denoed ρ 0 n, hen P, [a] nn D ρ 0 np, [a] n. 54 From 54, he aocaon probably follow mmedaely, becaue P [a] P, [a] / P. We defne γn D, q ρ 0 n! n n n nn D! qn n D, 55 D wre P [a] γn D, q P n p 0 x! S J D 56 PDxL x Q Dx dx., a S\a S A.4 Poon pror cae We wll now re-wre 56 a a conan leadng facor.e., a quany ndependen of [a] me a produc over only hoe arge ha produce a lea wo rack, J + D { J : a }. 57 Such a form allow he ue of effcen algorhm uch a JVC [6] o fnd he aocaon wh maxmal probably n he wo-enor cae. The fr ep o re-wre 56 o have a conan leadng facor. The leadng facor n 56 depend on n D, whch n urn depend on [a]. One can accomplh h by aumng a Poon pror ρ 0 n e ν ν n/ n! on he number of arge, where ν he pror expeced number of arge. Indeed, one canno ge a conan leadng facor whou h aumpon [4]. In h cae, 55 mplfe o γn D, q e q ν ν n D, whch allow 56 o be wren a e q ν P [a] P S PDxL x, a n ν p 0 x! J D Q Dx dx. S\a S 58 The econd ep o dvde each de of 58 by P [a 0 ], where [a 0 ] he null aocaon, whch map every arge n a e JD 0 o a o a cluer conanng only a ngle rack,.e., a 0 for all JD 0. The conan leadng facor cancel, we are lef wh he rao of a produc over J D a produc over JD 0. We mplfy h rao a follow. For each cluer of rack α [a] we collec he facor of P [a 0 ] correpondng o he rack n α.e., hoe wh JD 0 uch ha a 0 α o defne ν p 0 x PDxL R α x Q Dx dx, α S\α S ν p 0 xpdxl x 59 Q Dx dx, α S\{} where α S he e of enor occurrng n he cluer α. The produc of hee R α over all α [a] herefore he rao P [a] / P [a 0 ]. Becaue R α when α conan only a ngle rack, we may om uch cluer o wre P [a] P [a 0 ] R a. 60 J + D In he wo-enor cae, any cluer of a lea wo rack may be expreed n he form α {,,, }, 59 may be wren R α P,, 6 P where P P, νp p 0 xp DxQ 3 D xl x dx, 6 p 0 xpdxp DxL x L x dx. 63 Equaon 6 63 are he arng pon of he dervaon n he man body of he paper, hough hey are wren a lghly dfferen form here o faclae furher manpulaon.