A Bayesian Approach To Colour Term Semantics

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1 A Baya Aoac To olou T Satc Mk Dowa Mk@c.uy.u.au Ba Datt of out Scc, F09, Uvty of Syy NSW006 Autala Abtact A Baya coutatoal ol cb, wc abl to la t ag of bac colou t fo otv xal. Exal of colou a by atcula colou t a to a coctual colou ac, a Baya fc u to t t obablty tat ot colou co wt t ag of ac colou t. A fuzzy t ba otato fo ac colou t ca b cat by calculatg t obablty tat ac ot t colou ac co wt t xto of ac colou t. T la catgo ow t ototy tuctu caacttc of colou t, wt t bg a gl bt xal of t catgoy, agal b of t catgoy, a wt tat colou bg b of t catgoy to a gat o l xtt. T aoac a t avatag ov vou aoac of bg bot flxbl, a o bg abl to accout fo t full ag of obv colou t yt, but alo of ovg a c a xlct accout of colou t atc. T ol abl to accout ot oly fo t ag of colou t, but alo fo t acquto of to ag. Fut wok a to cooat at ba to t ol, o tat t abl to accout fo obv tyologcal att, a t lag ba of ua ubjct. Ky wo: bac colou t, ototy catgo, atc acquto, Baya fc, fuzzy t. Itoucto T a cb a totcal ooal about t atu of bac colou t. T ooal coc ot oly t atu of a ak kowlg of t atc of uc wo, but alo ow a o ay la t ag of uc wo. T a b uc wok o t ubjct of uc wo bot lgutc a ycology, a wll a lat cl, but t t wok a t avatag of bot bg abl to accout fo a w ag of colou t yt, wl ovg a xlct accout of t acquto of bac colou t wc a b lt a a coutatoal ol. Evy laguag a wo fo colou, but fft laguag wo ot fft ag of colou, o tat fo a colou t o laguag t wll ot alway b a wo aot laguag wt xactly t a ag. Bl a Kay 969 tf a ubt of colou wo wc ty call bac colou t. T wo a colou t wc a oooc, a wo xto ot clu tat of ay ot colou t. Ty ut ot b tct to aly oly to a aow cla of objct, a ut b ycologcally alt tat, ty ut b ot wt t ag of colou t kow by a vual, a b kow by all ak of a laguag. Bl a Kay t fo a al of 98 laguag tat all laguag av btw two a lv uc wo altoug t obl tat o laguag av twlv, a t a o agal xal.

2 . T Natu of olou It cla tat t xto of a colou t o ot cot of objct wt jut o cfc colou, but fact xt to objct wc av ay o of a ag of la colou. So o to uta t otato of colou t, w ut co t atu of colou, o tat w ay t ow uc a ag of colou coul b cf. Ftly, w ay co t ycal ot of lgt, a ow t ot vay btw lgt of fft colou. Lgt wav of fft colou av fft wavlgt. R lgt a t logt wavlgt, wt oag, yllow g, blu a ul avg cagly ot wavlgt. owv, lgt ca b a u of a xtu of t colou, a ca occu at vayg tt. c t cv oologcal colou of lgt o wc wavlgt a t, a o t tty of ac wavlgt. olou fo a cotuou ag wt at o xt a blu at t ot Too, 995. owv, t fact tat lgt a uc a ycal tuctu, o ot a tat t t fo wc t vou yt oc colou. I fact, oly t ty of cto cll av b tf t ta of t y, a ac ty o fftally to lgt of a gv wavlgt. I a fg to t cll kow a co; t ta alo cota cll call o wc a blv ot to lay a otat at colou vo. Eac ty of cll axally actv w t wt lgt of a atcula wavlgt, a t actvty ca gaually a lgt a of gat o l wavlgt t. So, wl ycal lgt a t oty of avg a cotuou ag of wavlgt, t y oly colou t of ow clo t fqucy of lgt acg t y to t focal fqucy of ac of t t ty of cll, a ow t t Too, 995. Fut ac a tac t uoyologcal atway of foato about t colou of lgt a t oc by t vou yt, a a tf a atcula cla of cll kow a oot cll. T cll oc t outut of t co to ouc a gal ba o a cobato of t outut of t cll. So cll oo g a lgt, a o o ot t c of g a abc of wavlgt. Ot cll alo oo g a lgt, but av oot o, og ot t c of wavlgt a abc of g o. Slaly, t a two ty of cll wc oo blu a yllow lgt, ac ty og ot t c of wavlgt of o colou a t abc of wavlgt of t ot colou. T cll c a t ut lgt oto a two oal colou ac, a ow Fgu. T colo of lgt wll b t by t g of blu o yllow, coog to t ozotal ax, a t g of o g, coog to t vtcal ax. A t k of oot cll a alo b otulat, wc woul oo lgt a ak lgt, a o cat a t o of lgt Kay a McDal, 978.

3 blu yllow g Fgu. T Two Doal olou Sac So fa I av cu t atu of colou fo t ctv of t ycal tuctu, a uoyologcal vc a to ow ol oc t, but w ay alo co ycologcal a oologcal vc a to t atu of colou. Wl ycally a t logt wavlgt of ay colou of vbl lgt, a ul t ott, oologcally a ul a la colou, a t a o colou wc coul b co a agal xal of bot a ul. T a b uc ac o colou fo ycologcal a oologcal ctv, a gal cou a b ac tat, oologcally, colou a a t oal tuctu, wc o cloly lat to t ot of oot cll ta to bot t ycal tuctu of lgt, a to al tag of uoyologcal ocg Too, 995. T t o u cbg t oologcal tuctu of colou a u, atuato, a lgt. Ftly, lgt coo to ow bgt o ak a colou, a coo cloly to t otulat lgt a ak oot cll. T o of u tf t oty of lgt wc cag cotuouly fo to oag, yllow, g, blu a t ul, a fally back to aga. T o otogoal to tat of lgt, a alo ccula, a f o og alog t o cto, o att w t tat ot, t ogal u wll vtually b ac. u coo atly to t two oal uoyologcal colou ac t Fgu, toug tat colou ac alo coa t oologcal o of atuato. T o of atuato coo to ow vv, o ulut a atcula colou, ctv of t u. At a lvl of zo atuato, all colou a black, wt o gy, vayg oly o t lgt o, ot o t o of u. At all ot lvl of atuato, a colou ay av ay u, wt atuato acg a axu w colou a at t ot vv, fo a gv u a lgt. T two o a ow Fgu, wc llutat ow atuato ay b co a aal o, a u a ccula o. T o of lgt otogoal to bot t o. 3

4 ul oag u blu atuato yllow tuquo yllow-g g Fgu. T oologcal Do of Satuato a u T t oal colou tuctu ba o t o of u, atuato a lgt coo to ow ol cb t xc of colou, a ow ty catgo colou. It alo aa to b t colou tuctu wc allow t bt a ot coc cto of t xto of colou t, a wc a c a b aot a t yt u alot all vtgato to bac colou t. I ow ov o to co a ub of totcal aoac to t atc of uc wo, ay of wc a uc coo wt t aoac u t a, but all of wc av at lat o otat ffc.. Aoac to olou T Satc Oft laguag atc a xla ug a fatu-ba aoac. T ag of a wo wll b xla t of cay a uffct fatu. So t ag of a wo uc a baclo woul b coo to t fatu, a a ua, wc togt a bot cay a uffct fo tg t oty of bg a baclo. T fatu ay tlv b coo to o tv lt, fo xal a coul b coo to two-lgg a aal. T aoac faly uccful at catug t ag of wo uc a baclo, a t a cla tcto btw tg t wol wc a baclo, a to wc a ot. owv, t ffcult to ow uc a aoac coul b uful utag t ag of wo uc a colou t. Taylo 989 ot tat colou t a a xal of ototy catgo. Fo ay colou t, ay Egl blu, ak ca aly tfy a gl bt xal colou fo t t, wat call a ototy. olou la to t bt xla wll b co goo o ba xal of t colou t to a vayg g, g o ow la ty a to t ototy. olou wc a o la to t ototy a co wo xal of t colou t ta to wc a o la. laly, a ot wll b ac w a colou o la to t ototy tat t o log a xal of t colou t at all, a ta co wt t otato of a gboug colou. owv, colou t alo xbt aot oty tycal of ototy catgo, tat t a o xal about wc t ffcult to t wt ty co wt t otato of t t at all. T wll b colou towa t y of t catgoy, a t oft agt btw ak about wc colou t t coct o to a uc colou. tal to ototy toy Taylo, 989, t a tat t ototy lay a ol tg wc objct o colou co wt a catgoy, a wc o ot. owv, xactly ow t 4

5 ototy f t xtt of t catgoy uually ot a xlct. obably t ot xlct ototy aoac to colou t atc tat of La 994. La f a colou catgoy by cfyg t catgoy ototy, a t z of t catgoy. T g of b of a colou t catgoy o t tac fo t focu, a t z of t catgoy t by a uc aat cfyg ow aly t g of b t catgoy ca a t tac fo t focu ca. La alo oo tat ol ca fo t ba of a accout of ow t ag of colou t ca b la. owv, I tk t ct t a o obl wt ol, wc alo av lcato fo wt t ototy aoac t coct way to f colou catgo. La ol o ot cfy ow a laguag la ca tabl t foc of colou catgo. Ita t au tat t ut b fx, uably atly. 43, a lag wll t cot of tg t xtt of t colou catgoy. c, t aat ajut ug lag ly ajut t z of t catgoy, a ca t affct t locato of t focu, o t a. T owat oblatc, a tyologcal vc to uggt tat ot all catgo av uvally t foc, a t a of catgo, a c xactly w t boua a, ctaly va btw laguag. Fo t ao I blv tat La wok lluat o of t ot oblatc act of t ototy aoac to lgutc catgo. I a awa of oly o ot coutatoal ol of colou t atc, a tat t wok of okla 997. okla u lf-ogag a, wc ca b a a fo of ual twok, to la t ag of colou t fo xal. okla ol otvat atly by uoycologcal vc, but t tak lttl accout of t cfc ltatu o colou cto a colou t. Ita of bag t ut to t ol o t oologcal o of u, atuato a bgt, t ut ba o, g a blu coot of colou, uc a a u by cout gatg colou fo lay o a oto. c t ulkly tat okla ol wll accout atculaly wll fo ycologcal a lgutc ata o colou t. owv, t tat lf-ogag a ay wll ov uful t coutatoal ollg of colou t atc, a ov a ttg altatv bot to ototy aoac, a to t aoac tak t a. Ot ta coutatoal aoac, a t ot xlct aoac to colou t atc tat of Kay a McDal 978. Ita of ug a tctly ototy aoac, uc a tat of La, ty gv colou catgo a ttato t of fuzzy t. Fuzzy t a t w t lt t t ca b b to a gat o l g, vayg fo o full b to zo ot a b of t t at all.. A factoal ub btw o a zo cat tat a lt a b of t t to a tat g. I Kay a McDal aoac, t ototy of a colou catgoy gv a b valu of o, a t g of b t t ca t fut away a colou fo t ototy. T b fucto, wc ak at t ototy, a ca o t, a ba o t o fucto of oot oc cll t ta of t y. A t catgoy b fucto, a c t catgoy ototy, a t atly by ual o fucto, t lttl oo fo co-lgutc vaato, o fact of lag of catgoy tuctu at all. Wl Kay a McDal oo tat coot catgo ca b v fo t fuzzy uo of t t f by t o fucto of o ta o ty of oot cll, a o-ay catgo uc a bow a ul fo t fuzzy tcto of t o I oul ot tat t ay uo of okla wok wa coc wt laguag tcology, a ot cogtv olg, o t uug tat t o ot tak accout of ycologcal a uoyologcal ata. 5

6 fucto, otw t colou catgo a uvally t by t o fucto of t oot cll. c Kay a McDal aoac oblatc, a t locato of t foc of coay colou t, a of t boua of all colou t, aa to vay btw laguag, a o uably ut b t, at lat at by lag. Alo, t o ot to b ay ao wy t g of b a colou catgoy oul lat ay ct way to t ual o fucto of cll t ta, a o a ol ba o t auto gt b co owat ulkly a o. T ol t t a alo ak u of t fuzzy t aoac, but t a of t b fucto of t t fg t otato of t colou t t by lag fo xal, a ot fo at ual o. 3 Baya Aoac to ycology a Lgutc T coutatoal ol u t a u a fo of Baya fc to la atc tato fo colou t. I t cto, I wll jutfy wy t laubl tat uc a ol ay gv a accuat accout of ow ol la t ag of bac colou t. T ay ao fo uog tat ol ay u Baya fc to la bac colou t tat t a ay ot ycologcal oa wc aa to b bt xla ug Baya fc. Ftly, Bt a atwgt 997 av u a Mu Dcto Lgt ba ol to accout fo ow cl ca ultaouly la to gt c, a to bg t oc of lxcal acquto. a ow tat uc a aoac coul allow cl to ak a og t tak, a o t laubl tat ty u o uc tatgy to a lag to gt c a acqu a lxco. Mu Dcto Lgt ca b a a vo of Baya fc, a o uc aoac uggt tat Baya ol ay wll gv a goo accout of ot act of ycology. uttloc, g a Vva 000 ow tat, at lat o ccutac, ol aa to u Baya fc to ov accuacy jugg xactly t tul. Dowa 000 a ow ow Baya ba fc t ca Mu Dcto Lgt ca a t acquto of ytax, a atcula ca accout fo t acquto of t ubcatgozato of ay vb. Fally, Ello 99 a ow ow la lag algot ca accout fo t acquto of ay act of oology. Altoug t abov a vy bf ovvw of o of t wok ycology a lgutc wc ak u of Bay to, t lcato of uc wok tat ay ycologcal oc ay av a Baya ba, a o t lkly tat t wll alo b tu of t acquto of bac colou t. c t uggt tat t wotwl vtgatg wat ot bac colou t woul av f ty w la ug Baya fc. If a Baya ol of colou t atc coo wll wt obvato a cocg colou t, t t wll b goo vc tat ol o la colou t atc ug o fo of Baya fc. I ow wat to cb a lttl o tal o a by Tbau a Xu 000, a aot by Gfft a Tbau 000, a t two a a cloly lat to aoac tak t a. Tbau a Xu, coo wt t aoac of t a, u Baya fc to ol t acquto of wo ag. T ol ct tat t ag tat ol woul attbut to a wo woul o t ub a ty of xal of t u ty a obv, wc coo wll wt cal fg. Fo xal, t ol ct tat f a o obv a wo bg u to f to a og, a tat wa t oly xal of t u of tat wo wc ty obv, t ty woul b uu wt t xto of t wo coo to t t of all og, t t of all aal, o jut t atcula og f to. owv, aft g 6

7 val xal, all of wc w og, t ol wa alot cta tat t wo xto wa t t of og. Altatvly, aft g xal cotg of t fft k of aal, t ol woul b alot cta tat t wo ot t t of aal, o aft g val fft xal of t a og, t ol woul bco alot cta tat t wo ot oly t atcula og. T wok otat ow Baya fc ay b u to la wo ag fo oly otv xal of t u of a wo, a Tbau a Xu w alo abl to otat tat t galzato a by t ol w vy la to to a by ol w t wt t a vc. T Baya ol wc ot cloly lat to t wok t a, tat of Gfft a Tbau 000, altoug t wok ot tlf coc wt wo ag, o wt laguag at all. Gfft a Tbau vtgat ow ol ca ct t fqucy wt wc o vt occu, ba o obvato of ow log c t lat occu. Fo xal, f ol a tol tat o aval at a ubway tato t a b 03 co c t lat ta av, t ty wll gu tat t ot lkly tat ta u vy fw ut, cott wt t cto of t Baya ol. owv, f ty a t tol tat o two ubqut vt to t ubway tato t a b 34 co, a t 7 co c t lat ta av, t ty a lkly to blv tat ta u wt a fqucy uc clo to 03 co. Aga t cott wt t cto of t Baya ol. Wt t ol, yot coo to ow oft ta av at t tato. T gl ot lkly yot wll b tat wc lag oug to clu jut t aval t of all t ta, but t ol f t t uc tat t qually lkly tat t tu tval btw ta gat ta o l ta t t. c t valu f fo t lgt of t btw w ta av, wll alway b gat ta t logt obv t la c a ta av. owv, a o ata obv, t f tval wll gt clo to t gt obv t, a wt o obvato t o lkly tat at lat o of t obv t co clo to t axu t tat ay la btw aval of ta. T ao tat t aoac to colou t atc t la to Gfft a Tbau aoac to fg t fqucy of vt, tat bot colou a t ca av cotuouly vayg valu. c, w ca u uc cal to t o t colou ac, uc t a way a Gfft a Tbau u uc a cal to t t. T followg cto cb t ol of colou t atc tal. 4 A Baya Mol of t Acquto of olou T Satc T coutatoal ol oo a accout of ow t atc of colou t ay b la fo a ft ub of xal. It au tat, ug t t w a o lag t laguag, ty wll obv a ft ub of xal of t u of a colou t, a wll b abl to t t colou of t objct f to ug t colou t o at lat o of t occao. It t occao, w a laguag la atc a colou t to a xal colou a by tat t, wc ov t ut to t oc of acquto. c t ata fo wc t ol wll la t ag of colou t a t of xal of cfc colou, a ac uc xal wll b a wt a wo tougt to ot tat colou. 4. -Lgutc oc a Rtato Bfo t obl to oo a ol of t acquto of t ag of colou t, t ft cay to t wat fo t -lgutc ut to t oc of acquto wll tak. laly ol ut b abl to cv colou bfo ty a abl to la t ag of colou 7

8 t wc ot ag of colou. c, o fo a o to acqu t ag of colou t, t ut b oc wc, takg a t ut t ycal lgt wav tg t y, oc t gal to fo a tato of t colou wc ay v a ut to lgutc oc 3. A ot abov, t lato btw t ycal ot of lgt tg t y, a t cv colou ot l, a ut b oat by a ub of tvg oc. Muc kow about t yology of t colou yt, a t a a ub of to to accout fo oa uc a cv colou cotacy t vayg lluato fo xal La 977. owv, t a ot coc wt uc oc; t ly au tat uc oc xt, a tat ty a abl to a fo ycal colou to a tato coog to oologcal colou. A ot abov, oologcally colou a a t oal tuctu, vayg o t o of u, atuato a lgt. At t, t acqutoal ol coc oly wt t u o. If oly colou of axu atuato, a t g of lgt at wc axu atuato ay b acv a co, t t colou wll b aag a o oal colou ac a ow Fgu 3. If, tatg fo ay gv u, t u atly cag to a gboug u a cott cto, vtually t tal u wll b tu to aft gog toug all t ot u fo t co atuato a bgt. ul oag blu yllow tuquo yllow-g g Fgu 3. T oologcal olou Sac T ol au tat o to laguag lag uc a colou ac avalabl to a laguag la, a o ty wll kow at lat ubcocouly tat, fo xal, o la to ul ta t to blu. laly, colou t ot colou coog to t oal volu of t colou ac, a ot ly to at of a o oal colou ac. It c a auto of t It oul b ot tat a xcto to t t acquto of colou t by cogtally bl cl. Laau a Glta 985; ct Bloo, 000 fou tat bl cl kowlg of colou wo wa ay ct la to tat of gt cl of t a ag. Fo xal ty w awa tat colou wo blog to a gl oa, a tat ty aly oly to coct objct, a wll a bg awa tat ty ot a oty wc t cl tlv coul ot tfy. Wl t tat t ccutac ay act of t ag of colou t ay b acqu, t ot ctal act of t ag of colou t, tat t otato of cta ag of colou, caot b acqu uc ccutac. T vc o owv uggt tat cotxtual a ooytactc cu ay b a atoal ouc of vc u by cl tg t ag of colou t, altoug uc cu a ot at t utlz by t ol oo. 3 Fo t uo t au tat uc oc a lac bfo laguag lag bg. It of cou obl tat gfcat vlot of uc oc ay tak lac ug t acquto of t ag of colou t, but t au tat t ulkly tat uc ffct wll av a ajo act o colou t acquto, a o ay b go. 8

9 t ol tat colou t acquto ay b ufully oll w cog oly a gl o of t colou ac. T auto a fo t uo of lcty oly, a t o ao to uo t ol oul ot b xtbl to oat ov t full t oal colou ac. owv, t tcto a tat t ol wll b coc oly wt t coatc colou t. T ag of acoatc t uc a black, wt a gay caot b la wt t t ol, a o ca t ol tgu btw colou t wc ff cally o t o of lgt, fo xal a k. T ol, owv, abl to gv a accout of t acquto of act of ag t by ffc o t o of u, uc a to btw, oag, yllow, g, blu, a ul, a ay ot t ot laguag. W a la obv a colou t xal, t wll b t a a ot t oologcal colou ac, a t ot wll b labll wt t colou t tlf. Fo t uo of t ol, u wll b ub ug a abtay ubg c, wt t og zo bg t ac, a cag toug oag, yllow, g, blu a ul, u to u 99, wc wll aga b t ac, ajact to u 0. Ug t c, t obl to t tac of obvato of colou t ly by a ag of t colou t a t coog u ub. 4. Galzato fo Exal to Ot olou So fa, t fo of t ol coctual colou ac, a t tato of obv colou a b cb. owv, t o ot a t u of ow ol wll ag a to vouly u colou, w to colou a obv t abc of a lgutc labl. T ol u Baya fc to t ow lkly t tat a gv colou t a coct labl fo a atcula colou, galg fo t xal of t u of tat colou t alay b by t ol. Wt t cotxt of Baya fc, a yot wll b a cfcato of wc colou ca b coctly labll wt a gv colou t, a t obv ata wll b t t of xal colou t wc t ol a b avg b labll wt t colou t u coato. Bay to, a gv wll b u to t ow lkly ac gv yot gv t obv ata, but to o t t ft cay to cfy wat t obl yot a, a ow lkly ac o a o, a ow to t ow lkly t tat ay gv t of colou wll b obv to av b labll wt t colou t quto, w aug tat a cfc yot coct. 4.. obl yot a t a o obablt yot a to t otato of a colou t all coo to a cotuou cto of t oologcal colou ac, a ow Fgu 4 blow. Eac yot wll av a tat ot,, a a ot,. A yot tat tat a colou a coct labl fo all a oly to colou wc fall aft t tat ot, a bfo t ot, w t colou ac follow clockw. Eac of t ot ca b fc ug t ubg c fo t oologcal colou ac cb abov. c, t z of t colou ac ot by a colou t coog to a atcula yot wll b gv by -. I t ca w t ag of t colou t coa t og, t 00 t z of t oologcal colou 9

10 ac ut b a to, a t ca t valu of woul otw b l ta, ultg a gatv valu fo t z of t colou ac ot by tat t. 0 xtt of colou t tat Fgu 4. A yot a to t Dotato of a olou T A yot ay bg ayw wt t oologcal colou ac, a ay ayw wt t ac. All obl tat ot a co qually lkly a o, a a all obl ot gv ay atcula tat ot. c, t wll b a cotuou ac of yot, a yot wll ag z fo ot clug ay of t colou ac at all, to clug t wol of t colou ac, wt all z of yot bg co qually lkly a o. 4.. T obablty of Data gv a yot Gv t abov cfcato of yot, t ow cay to cfy ow obabl t tat t b colou xal labll by t cut colou t woul av b obv gv a atcula yot. T wll cfy ow to calculat t t fo quato abov. It au ft tat laguag la ak o a o auto tat cta colou a o lkly to b a by colou t ta a ot colou. T auto obably ot tly accuat, a cta colou a alot ctaly o fqut a o vot ta ot, a lgutc fc to o colou lkly to fulfl gat uo ta fc to ot, a o fo t ao w woul ot xct all colou to b a wt qual lkloo. owv, a t t ulato all colou a qually lkly to b a, t a aoabl auto to ak. I a vot w t wa ot t ca, t tat la woul av to tak accout of t facto, a ajut t lag tatgy aoatly. taly t to b o ao wy t ol coul ot b xt to tak accout of uc facto, but fo t uo t a colxty ucay.. Fut u coc ow a gv colou xal co to b a wt a gv colou t. Suc xal woul b v aly by a o obvg t u of a colou t, a fg t tty, t colou of wc t colou t wa bg u to tfy. T la woul t a t colou t wt t cv colou, a b t a fo lat fc. 0

11 4... Lag wt oltly Rlabl Iut T lt auto wc w ay ak about ow a la vw t ouc of vc a to t otato of a colou t, tat ty ay alway co t ag of t colou t a t colou to b coltly accuat. Gv t auto, t obablty of ay colou xal fo t colou t u coato gv by 4 fo ay yot wc clu tat colou t ag. T yot ly tat tat a xal of t colou t qually lkly to occu ayw wt t ag of t yot. c t obablty of a gv colou xal occug at ay atcula ot wt t ag of t yot qual to o v by t ub of uc ot -. I ot ca t wll b o ta a gl xal of ac colou t, o t obablty of all t xal wll b qual to t ouct of t obablty of ac, o log a ac wt t co of t yot. A t obablty of ac vual colou xal gv t yot gv by, t obablty of all t ata gv a yot gv by 3 5, w t ub of xal of t colou t. 3 owv, t abov quato oly al t ca tat all t obv colou xal a wt t ag of t yot. A t auto a b a tat all t obv colou a coct xal of t colou t, t obablty of ay vual colou xal bg out of t ag of t yot zo. c t obablty of t ata gv a yot wc o o o colou t xal gv a coct labl gv by Lag wt Ulabl Iut It wot cog wt t abov auto, tat a o wll alway au tat a gv colou t a colou xal wll alway av b coctly a, coct. T a goo ao to blv tat a laguag la woul ot b coltly cta tat ty a alway coctly tf t colou bg f to by a atcula colou t. T ag of a colou t a a colou l o coctly tfyg t colou t bg u, a tablg wat colou t ak t to tfy ug t t. T a a ub of facto wc coul la to a o coctly ag a colou t a a colou, catg coabl co fo o vg colou t a colou a. Ftly, a ak ay u t wog colou t to cb a colou wc ty w to f to. Suc a o coul a fo a ub of ao. O o occao t k of o woul a 4 Not tat t quato au tat colou a cf wt a accuacy coog to t ut z of t cal of u u. owv, f colou a cf to a fft accuacy, t wll ot affct t cfcato of t ol, a blow w a foula fo t fal atc ttato of a colou t cf, t coot v by t tgal of t t ov all yot, a o ay cotat t calg fo t co wt wc colou a co woul cacl out. 5 T foula au tat t ub of colou xal kow a o. A t valu o ot cag btw yot, t auto wll ot av ay ffct o t ol.

12 ly toug a c o, o t ak ay av coctly cv t colou. T oblty woul b cally lkly u oo o uuual lluato, a t coto woul ak t tak of accuatly cvg t colou o ffcult, toug v all o cvg colou fo colou a t ag of t otato of colou t coul ouc t k of o. owv, v f a ak cv a colou accuatly, a o ot ak a foac o, ty ay tll labl a colou t accuatly. T woul a o of two ca, t bcau a ak wa uu of t coct colou t wt wc to labl t colou quto, o bcau ty a coctly la t otato of t colou t. If a ak a obv oly a lt ub of xal of a colou t, t ty a ulkly to b cta about xactly wc colou t t ca b coctly u to f to, a t cally lkly to affct colou a t ag of t otato. It alo obl tat a ak a ly la t otato of a colou t accuatly. T coul a fo a ub of ao, clug t ak avg la fo accuat ut. Et of t two facto coul la to a la coctly ag colou t wt colou. T abov cuo ay to uo a gl coct otato fo all colou t. owv, t ay oblatc, a xactly wat a coct otato coul b, ot ta t otato blv to b coct by a ak of t laguag, ucla. It o aoabl, owv, to co t to b a coct otato, ba ot o t atc tato of a vual ak of a laguag, but o o k of o of t c couty a a wol. I wll ot cu xactly ow uc a c couty o ay a fo t kowlg of vual ak. Ita I wll ly au tat actc vual u of colou t to a colou ay b claf a coct o coct, ctv of t laguag u of ay vual ak. avg co t ca wc a ak ay co to u a coct colou t to cb a atcula xal colou, t ow a to co o wc ay a v ca wc a ak a u a colou t coctly. Eo coul a toug t la blvg tat t ak a u a colou t to tfy a colou, w fact o wo a b u to ot a colou, w a a coctly f t colou wc t colou t wa u to cb, o t a ay ly av cv t colou. Ftly, a a coul co to blv a colou t a b u to cb a colou, w fact t a ot b u to cb a colou at all. olou wo a uually olyou, a ay alo av ooy. o, fo xal, t tc May a g fg. A laguag la gt t f tat t t g ay b u to cb t colou of May fg, w fact t ak at to ay tat May vy goo at gag. Eo of t ot coul alo a f t la cv a wo wc ot a colou t, uc a way tat ty blv tat t. Ty a t lkly to aocat a colou wt t wo ty blv to av b utt, w fact tat wo a ot b u by t ak. Suc o a cally lkly gv tat acquto of ot aa of laguag takg lac ultaouly to t acquto of t atc of colou t. Scoly, a a ay cv a colou xactly t a way a a ak gt, wt t a coquc fo lag. owv, obly t ot otat facto lag to a la cog to a a colou t wt a colou xal coctly, tat ty ay tfy t objct, o t atcula at of tat objct, t colou of wc wa bg f to. Dtg wat colou a colou t wa u to cat o a atcula occao of u a colx oc, wc a laguag la ut ak fc about xactly wat objct t tat a t colou wc ot by t t t ak ug. Muc a b wtt about t oc fo

13 xal Bloo 000 cu t at lgt, but t coutatoal ol cu wll ot focu o t act of wo lag. Fo t uo t uffct to ot tat t a colx oc, a t a ay ootut fo o. t wll ly b au tat colou xal wc o ot co wt t otato of a atcula colou t ay otl o o occao co to b a wt tat t. Suc oou ag ay a fo ay of t ao to abov, a obly alo fo ot ao ot co. Gv all of t abov coato, t cla tat a la wll ot b coltly cta tat all ca t colou wc ty blv to av b a wt a atcula colou t, actually a colou tat coctly fall wt tat t otato. c foula 3 a 4 abov o ot coctly cfy ow oul b calculat, a bot t foula au tat t laguag la blv t ut ata to t oc of galato fo xal to b coltly accuat. It cay to cooat to t foula a facto accoutg fo t oblty tat o of t xal colou fo t colou t u coato a lkly to b coct. Fo t ao a cotat touc, wc coo to a o blf tat a colou a b coctly aocat wt a colou t, a oo to t colou abtaly bg ayw wt t oologcal colou ac. Mo cfcally, t obablty wt wc t la au tat ac colou xal a b coctly a wt t colou t labl, a t obablty wt wc t la au tat t al labl coltly abtay, a c a ay o ay ot coo to a coct colou t fo t colou quto. I t ca w qual to o, t ol wll coo to t ca outl abov fo colt ctaty of coct colou a colou t ag. T tatt of a la cofc t coct of t ata of cou vy l. It au tat fo all colou t, a fo all xal colou, t wll b a qual obablty of oouly atcg a colou a colou t. owv, t a a ub of facto wc gt affct t cofc wc a o a about coctly tfyg t uag of a colou t. Ftly t ccutac wc t colou t w atc to atcula colou gt la a la to b o cta tat ty a coctly a a colou t wt t colou t wa u to f to o o occao ta o ot. T woul ly tat fft valu of t cotat gt b aoat fo fft colou xal. ol gt alo ajut t cofc tat ty a abl to accuatly a colou t to colou, by cog ow wll t f xal of ac colou t coo to laubl otato. Suc coato woul gv a o o cato of ow obabl t tat ty a b uccful ag ac colou t to t colou t wa u to ot. I t ol t woul coo to ajutg t valu of t aat to a aoat valu takg accout of t obv ata. T ffct of t facto coul b vtgat xto of t ol, but fo t uo a o cofc t coct of t ata oll ly by aug a fx valu fo t aat. Aot auto of t cut ol tat, f a colou ot a coct xal of t colou t, t qually lkly to occu ayw t oologcal colou ac. Ev gog t oblty tat o colou a o fqutly a ta ot a cu abov, t auto ay ot b accuat. Wt t auto val woul to o ow t colou t a co to b a wt a coct xal colou. Fo xal, f t o ao toug t ak o a avg cv t colou, w woul xct t colou to b a 3

14 t boua of t otato of t colou t. I cotat, f t a a tf t objct to wc t colou t al, o t ak a b ug a colou t to ot a oty ot ta colou, t t auto woul jutf. c t xtt to wc t auto jutf t o t latv otac of facto uc a t. owv, ug t cou of t acquto of colou t, t woul to b a vy ffcult tak fo cl to a ow accuat t ata fo wc ty a lag. Oly w ty a tf t ag of t otato of ac colou t wt a g lvl of accuacy, woul cl b abl to ak accuat tat of ow lkly t tat t obvato of colou xal fo ac colou t w coct. c, t laubl tat ty ak l auto uc a to oo, about bot t lablty of t ut xal, a t ot of ay oou ata. taly t a altatv auto tat coul b a, ay of wc gt b qually laubl. oato of uc altatv woul b a wotwl avu fo ac, but at t t uffct to co oly t o oblty. Gv t aat fo cofc t ata, t ow cay to v w foula cfyg t obablty of t ata gv a atcula yot. Ft of all cog, oly a gl colou xal, f t kow tat t a b coctly a wt t yot, t, aug t yot to b coct a c alo tat t colou xal co wt t otato, t obablty gv by 5. It ay b ot tat t t a a quato abov fo coltly labl ata. 5 If t kow tat a colou xal wa ot coctly a wt a colou t, a c w au tat t qually lkly to occu ayw wt t colou ac, t obablty gv by owv, a la wll ot fact kow wt a colou a b coctly a wt t yot, o w ut cooat to t foula t obablty wt wc a la au tat t colou, o ot, a coct xal of t u of t colou t. Ftly w ca co t ca wc t colou xal out of t ag of t yot. I t ca t colou xal ut av b coctly a wt t yot, a o t obablty gv by 6, wgt by t obablty tat a abtay xal colou a b tf to coo to t colou t. T obablty, o t ultg foula tat gv If a colou xal co wt t ag of t yot, t t ay av b coctly a wt t colou t, wc ca t obablty woul b v fo 5, o t ay av b coctly a wt t colou t, but otl co wt t otato, wc ca t obablty woul b v fo 6. I fact, o to t t ovall obablty of t xal gv t yot, w u t obablty of t xal u bot of t coto, wgt by t obablt wt wc t la au t of t two oblt. T 4

15 obablt a a - ctvly, o t ovall obablty of uc a xal gv by I o to t t obablty of all t ata gv a atcula yot, t cay to ultly togt t obablty of ac vual xal. W t a xal wc fall wt t ag of t yot, a xal wc fall out of t ag of t yot, t obablty gv by T abov foula t o wc a b lt o a to allow t obablty of t obv ata gv ay of t bl yot to b t. owv, t ot ow obabl ata gv a yot tat w a tt, but fact t t obablty of yot gv t ata. I o to calculat t obablty ug Bay to w to co t obablty of o fut coot, a cb t followg cto T A o obablty of t Data T fal t o t gt a of Bay to a gv,, coog to t a o obablty of t ata. T t cf ow lkly t ata wa bfo w co ay atcula yot. Wl t t cotat aco all obl yot, w to co t o to t t obablty of ac vual yot gv t obv ata. T valu of t t ca b calculat by ug t obablty of t ata gv ac vual yot, ultl by t a o obablty of tat yot. A t yot a cotuou tat, t a o ct boua btw yot w ca u calculu to calculat t valu of t u. T x atatcally 0, w t t of all obl yot T obablty of a yot gv Obv Data W ow av a cfcato of all t t wc w to colt t gt a of Bay to a gv. Subttutg t xo gv 0 to t quato, w obta. It oul b ot tat o of t occuc of av b labll, a otw t woul b two fft vaabl bot gv t labl. 5

16 A all yot a qually lkly a o, t t a a bot cotat a qual to o aot, a c wll cacl, allowg t quato to b wtt a. T quato u t xt cto to t ow lkly t tat ac obl colou coctly labll wt t colou t u coato Galg to Nw olou T ol u Baya fc, ot to t t obablty of ay vual yot, but to t ow lkly t tat ay colou wc ay b of tt coctly labll by a atcula colou t tat ow lkly t tat t colou co wt t otato of t colou t. Fo a locato t colou ac, x, w ca x t obablty tat t wt t co of a colou t, gv tat w au tat yot coct, ug t xo x. W t locato x wt t ag of t yot, t t xo wll av a valu of o, a f a yot coct, t all u wt t ag co wt t otato of t colou t. owv, w t locato x out of t ag of t yot, t t xo wll av a valu of zo, a f t yot coct, t all locato out of t ag o ot co wt t otato of t coog colou t. owv, wat a to b o to v a fal xo fo t obablty tat a ot t colou ac wt t otato of a colou t gv, ot a atcula yot,, but gv all t obv ata,. I o to o t, t cay to u t taa Baya ocu of yot avagg. T obablty tat a locato t colou ac wt t otato of t colou t gv by ug t obablty of t gv ac vual yot ultl by t obablty of ac yot gv t ata. A wa ot abov, yot a cotuou, o calculu u to fo t uato, wc x atatcally 3, w aga t t of all obl yot. 3 x x T foula colt t cfcato of t Baya ol of acquto, by cfyg ow a o lag a laguag ca la to ct wc colou co wt t otato of a atcula colou t, w t oly foato about t a ft ub of xal of t u of a colou t Mo ta O olou T So fa t ol a b cb wt ct to oly a gl colou t. T bcau ac colou t wc a cl t to la co tly of all t ot. Evy xal colou wll b b oy xactly t a way, but ac wll b a wt a coog colou t. I cog t otato of ac colou t, accout wll b tak oly of to colou xal a wt t. T a tat, fo vy obl colou, a obablty ca b obta tat t wt t otato of ac colou t cout by t ol. c, t obl tat t ol wll ct tat a atcula colou vy lkly to co wt t otato of o ta o colou t, o tat t vy ulkly to co wt t otato of 6

17 ay colou t at all. owv, t ol wll v coclu wt abolut ctaty tat ay colou out of t ag of t otato of ay atcula colou t. owv, t otat to ot tat cog colou t o at a t a auto, a tat altatv lag tatg a obl. T ot obvou ofcato of t tatgy oo t ct, tat a la gt ao tat f a atcula at of t colou ac co wt t otato of o colou, t t l lkly to co wt t otato of aot colou. Fo xal, f a la w to blv tat a colou ca wt t otato of a t tok, ty gt t co t ulkly tat tat colou woul co wt t otato of ay ot t. A tog vo of t woul tat tat a la gt au tat t otato of bac colou t a atcula laguag o ot ovla at all, a o f a colou wt t otato of o colou t, t t wll ot co wt t otato of aot colou t. Blf of t k, tat t otato of colou t o ot ovla gt b uot by fc a fo t lgutc cotxt of a colou t. A la gt ak fc a to wt colou t ovla o aot, ba o agatc ttato of obv alog. Fo xal, f a la a a covato wc t ft ak ay I tat balloo ul? a a co o l No t blu., ty gt tak t a a cu tat t otato of a oag o ot ovla, a o co t otato of ac of t t w fg t otato of t ot. owv, v t ca t tat t auto ay b coct, a woul b t ca f t abov quto t ft ak a ak I tat balloo tuquo?, but cv t a ly. T bcau t otato of t colou t tuquo a blu ovla, wt o colou bg bot tuquo a blu. So t ca b tat ut of t ot ot labl ovg vc tat t otato of colou t o ot ovla. owv, t at lat a vy coo oty of laguag tat bac colou t a laguag ot oly o ot ovla, but alo atto 6 t colou ac. T u cu tal by Kay a Maff 999. T ao tat tuquo a blu ovla, bcau tuquo ot a bac colou t, but a coay colou t. T otato of coay colou t ca ovla wt tat of o ta o bac colou t, a t ca wt tuquo, wc ovla t otato of bot blu a g. Altatvly coay colou t ay ot a ag of colou wt t otato of a bac colou t, a t ca wt co, wc ot colou wt t co of t otato of. Wat t l tat, f a laguag la w abl to lably tfy wc colou t a laguag w bac colou t, ty gt b abl to la o ffctvly by akg t auto tat t full t of t t t laguag atto t colou ac. laly, lyg o cu of t ot a laubl lag tatgy, but t o a t ffculty of tg wc colou t a bac, wc woul to b ffcult, at lat utl t acquto of t colou t yt wa at a faly avac tag. c t ulkly tat aoto of uc a tatgy ca b of ajo otac ablg t uccful acquto of colou t. Fut, uc a lag tatgy t uabl to accout fo ow coay colou t a acqu, a coay colou t ctaly o ot atto t colou ac. Ev gat ffcult fo uc a aoac a cau by a all oty of laguag wc t bac colou t o ot aa to atto t colou ac, o colou avg o coog lgutc labl Kay a Maff, By atto t at tat t wol of t colou ac v u o tat ac colou ot by xactly o colou t. 7

18 4.3 Dvg Fuzzy St ug Baya Ifc A cb u to t t ot, t ol ly t t obablty tat a cfc colou co wt t otato of a atcula colou t. owv, o fa o coato a b gv to ow a atc tato of a colou t gt b v fo t Baya ol. owv, t Baya ol lctly f a fuzzy t tato fo t otato of colou t. Ita of cog t obablty tat vual colou a ot by a atcula colou t, w ca co, fo ac colou t oologcal colou ac, t obablty tat t wt t otato of t colou t. T wll ult a obablty fo ac colou tat t ca b ot by t colou t of tt, a t valu ca b tt a cfyg t g of b of t colou t atc catgoy labll by t colou t. c t valu ay b u to f fuzzy b t coog to ac colou t. T a a ub of ttg ot of t fuzzy t, ot obvouly tat fo ac t, a c fo ac colou t, o colou wll b b wt a gat ctaty ta ot colou. owv, t wll b a obablty aocat wt t b of vy colou vy t, o tat, wl t wll b co tat o colou a alot ctaly ot b of t t, t wll alway b a all obablty aocat wt t oblty tat ty a b. T lcato of t ot of t colou t otato cu tal blow, but ow I ov o to co ow t ol ay b actcably lt o a cout. 5 Iltato of t Mol Wl cto 4 cf t ol tal, t o ot cu ow t ol ca b lt actc. I atcula, quato a 3 bot cota tgato, but t a to b ow tat t lvat at of t quato ca fact b tgat actc, a ow t tgat quato ay b u to t ow lkly t tat ac colou of tt co wt t otato of ac colou t. 5. alculatg t obablty tat a olou wt t Dotato of a olou T If w ubttut t gt a of quato fo t t 3, w obta t quato 4. 4 x x T valu of t tgal of wt ct to ot t o t valu of, a o a cotat t t tgato wt ct to t vaabl. T allow quato 4 to b wtt a 5. 8

19 5 x x A t t x valuat to o t ca w x co wt t otato of t yot, a to zo ot ca, t to alf of t facto 5, ffctvly a u ov fo all ag of t ac of obl yot wc x co wt t otato of t yot. I cotat t t o t botto of 5 a u ov tougout t yot ac, gal of wt x co wt t otato of to yot o ot. c t quato ay b wtt a 6, w x coo to t u ov fo yot clug x t otato, a otx coo to t u ov t a t fo yot ot clug x t otato. T oga wll c lt 4 by calculatg x a otx, a ubttutg t valu to 6. 6 x x x otx 5. Dvato of t Itgal Now tat t tak of tg t obablty tat a colou co wt t otato of a colou t a b uc to tg valu of u of ov cfc ag of yot, a ubttutg t valu to quato 6, t cay to co ow t u ca b calculat actc. o aga quato 9, at a 7. T foula cota t ybol,,,, a. Wat t otat to t fo t oc of tgato, wt t valu a cotat aco fft yot,, o fo to t wc a vaabl, wat way ty wll cag Ft of all t ay b ot tat a cotat t tougout t ol. a coo to ow ay of t xal colou co wt t ag of t yot u coato, a ow ay out of t. c t valu wll vay btw yot, g o wc colou xal ac yot clu t ag, a wc t xclu. owv, t valu wll cag ctly at fx ot t yot ac, a o tgato ov t ac of yot caot clu yot fo wc t coog valu of a woul vay. Itgatg ov t wol of t yot ac woul oly b obl f a quato latg t valu of a to t yot coul b ubttut fo t valu. c t valu of t u ov wll av to b calculat cto, wt t valu of t u fo fft valu of a co aatly. At t ot t alo otg tat w w to v aatly t u ov fo to yot wc clu t ot of tt, x, a to wc o ot. T alo a oty wc wll cag ctly at ot t yot ac, a o laly t wll b cay to co u ov aa of t yot ac w x clu t ag of t yot aatly to to w x ot clu t ag of t yot. T fal two t to co a a, wc coo to t locato t yot ac of t tat a t of a yot. T valu f t atcula yot u 9

20 coato, a o w w u ov aa of t yot ac, w a fact ug ov quato 7 fo ag of t vaabl a. Rcall fo cto 4.. tat t yot ac,, coo of yot wc ay tat at ay ot t colou ac, a ay at ay ot. c, w w u ov aa of t colou ac, w ut u ov quato 7 fo a ag of valu of, a fo ac valu of, fo a ag of valu of. Rcall alo fo cto 4.. tat fo t uo of ltato t vaabl cfyg t of a yot,, wll alway av a valu gat ta tat cfyg t tat,. So o ca t valu of wll b gat ta t z of t colou ac 00, ca w t yot clu t og t ag. I fact, o ca w wll co cotuou ag of yot w t ag of tat valu alo co t og, a t ca t u lt of wll alo b gat ta 00, to cat a locato t colou ac clockw fo t og. A ug ov ag of t yot ac qu ug ov ag of two aat vaabl, t ut b lt ug a oubl tgato. 8 x ow t obablty of t ata gv a ag of yot t by ug ov a ag of t yot ac. owv t xo al quato, catg tat t u tak lac ov t full ag of yot, lac by two aat tgal cfyg tat t u b tak ov a cfc ag of yot, wc a t collctvly a. T cfc aa of t yot ac,, cota t ag of yot wc tat ayw btw t ot a, a wc ayw btw t ot a. Tougout t t of t a, I wll u to f to ay atcula ag of yot cutly u coato. 8 If w ubttut fo t xo 8, ug 7, w obta quato 9, cfyg xactly t tgato wc ut b fo o to v a quato allowg t u ov t obablt of a t of ata to b calculat fo a cfc ag of t yot ac. It ow cay to fo ft t tgato ov, a coly t tgato ov, o a to allow t xo to b valuat fo cfc valu of t aat Itgatg Ov W ca ot ft tat w wll oly u tgato to u ov aa of t yot ac wc a o ot cag, a c, fo t uo of tgato ov bot a, t t, alog wt, a all cotat. A t valu of ot t o t valu of, t too wll b a cotat fo t uo of t tgato ov. A a ft t fog t tgato, w ay ot tat a cotat t ay b ov tly fo t co of bot t tgato, o a to v quato Now t ay b ot tat t t to b tgat wt ct to a boal xo, o w ca u t boal xao, a gv to lac t t. Not tat t xao of 0

21 a cf by quato. If w quat a wt t ft at of t boal, a 3, a b wt t co at a 4, w ca obv t quvalc of t t to b tgat a t lft a of quato. b a b a 0!!! 3 a 4 00 b W w u xo to ubttut fo t aoat t 0, t ult quato By ovg cotat t fo t co of t tgato, a ovg t ct uato fo t co of t cotuou o, 5 ca b tafo to W ay ow ot tat t tgato to b fo tagtfowa, but tat t wll av a cal ca w - qual to o. T wll b t w ca qual to -, a o w wll u t ct uato u to oly, -, a t clu aat t fo w qual to - a w qual to. T ultg quato, aft t t av b aat out gv Slfyg t wc ow a ow a to t ow of o, o t ow of zo, ult quato 8.

22 W ut ot, owv, tat w t boal xao wa aat to t aat at, a lct auto wa a tat wa qual to o gat ta two, o tat lt of t uato coul b aat fo t ca w wa qual to o, a w wa qual to zo, a tat t woul tll lav at lat o gat valu of wc woul b cov wt t uato. owv, o ca wll b qual to o, o to zo, wc wll b o w t ag of yot u coato cota oly a gl colou xal, o o colou xal at all. Subttutg t valu of zo fo quato 6 ult quato 9, wc t quato to b tgat t ca tat t yot a o colou t xal Subttutg t valu of o fo quato 6, ult quato 30, wc t quato to b tgat t ca tat t yot a oly a gl colou t xal W ay ot tat quato 9 cota a tatato of t t coog to t ft tgal 8 a 30 cota a u of tatato of t ft a co tgal t a quato. c, at ta ocg wt t tgato of t quato aatly to tat of t quato fo w gat ta o qual to o, ty wll b v by ubttutg t aoat valu of to t t t fal tgat fo of quato 8. c w ow oc wt t tgato of t quato, ft wtg t fal tgal a 3, o a to ak t tgato o taat W t t tgato wt ct to a fo, t ult a gv 3.

23 [ ] [ l ] 00 0 T colt t tgato ov t vaabl. owv, bfo t tgato ov fo, t cay to co wat t valu of t lt o t tgato wll b, a t ult of t co tgato wll o wt a a cotat wt ct to t valu of. 5.. Itfyg t Lt o t Itgato T atu of t lt o t tgato wll b co wt fc to a xal of galg fo a cfc t of colou xal, to a vouly ulabl colou. Fgu 5 blow ow a tato of t oologcal colou ac o wc a ow t colou aocat wt t tac of t u of t colou t yllow, alog wt t ot, x, about wc t obablty of t bg wt t otato of t t yllow wll b calculat. x yllow yllow 3 yllow Fgu 5. T oologcal olou Sac wt Obv olou T Exal T valu of t t a wll ff g o ow ay xal of t colou t yllow a wt t co of t yot. c, w w co cotuou ag of t yot ac, t valu wll cag a t t oto of t tat o t of t yot bg co a o of t ot. Slaly, a t oto of t tat o t of t yot co t ot x, t oty of wt t yot clu o xclu t ot about wc w w to calculat t obablty of t cog wt t otato of yllow cag. A t tgal wll ot wok ov aa of t yot ac wc av fft valu fo t aat a, a w w to f aatly t obablty of t ata u ov all yot clug x, a all yot xclug x, w caot u t quato fo ag of t colou ac wc co t locato of a colou xal, o t ot x. owv, a w w to co t obablty of t ata gv all t yot, but, fo ao of ffccy a lcty, akg a fw calculato a obl, wt ac calculato, w wll alway co a lag a ag of yot a obl wt t cotat. I ot ca t wll 3

24 volv ttg t lt o t tgato to ot t colou ac t w t a tac of a xal of t colou t, o at t locato of t ot x. A a xal, w ca co calculatg t obablty of t ata gv all of t yot wc clu t xal colou labll yllow a yllow Fgu 5 abov, but wc xclu t ot x, a t xal labll yllow 3. T a ca of calculatg t valu of, w coo to t ag of yot. T yot a to t tat of wc a aft x, but bfo yllow, a wc av t aft yllow, but bfo yllow 3. c t valu o t lt o t tgal wll coo to t fou ot. T ft oto at wc a yot ay tat,, wll b t locato of x, a t lat oto at wc a yot ay tat,, wll b t locato of yllow. T ft oto at wc yot ay,, wll b yllow, a t lat oto at wc yot ay,, wll b yllow 3. W t valu a ubttut to t fal fo of t quato, t valuato wll t t valu of. I t abov xal, t lt o bot a w all cotat. T wll b t ca wv t at lat o colou xal o t ot x aatg t ag of oto wc t yot ay tat, fo t ag of oto wc t ay. T tgato uc ca colt cto owv t a o tac wc t coto o ot ol, o t followg aaga co tgato t ccutac. Lt u ow co aot xal, tat of calculatg w t t of yot u coato,, coo to a ca w t t a colou xal, o x aatg t ag of t tat of yot a t of yot. T wll b o wv w co yot tat cota all t colou xal a t ot x. If w co t wt ct to Fgu 5, t a fou aat ag of uc yot. To wc bot tat a btw x a yllow, btw yllow a yllow, btw yllow a yllow 3 a btw yllow 3 a x. og a a xal t ca of yot tatg a g btw x a yllow, t gt at ft tat t valu of woul b x a yllow, a tat woul b x00 a yllow woul b a to t latt valu to cat tat t yot woul go all t way aou t colou ac a at t og. owv, t cla tat t valu a ot coct w w co t yot wc tat at t alt obl ot, x, a f at t latt, yllow. T yot woul go all t way aou t colou ac fo x, but t lat cto, fo x to yllow, woul ovla tlf. T oblatc, a t yot lag ta t wol of t colou ac, a clu o of t colou twc, wc o ot av a cot ag wt t ol. Wat wog wt t abov valu o t lt of t tgato, tat ty o ot tak accout of t fact tat t of t yot ay ot aa o ta o full ccl aou t yot ac fo t tat. It ay to cooat t tcto to t lt o t tgato by ttg t u lt o t of t yot,, to 00, at ta to yllow 00. Now t of a yot ay ac all t way to yllow oly t ca tat t alo xactly w t yot tat. Icooatg t w lt o tgato to quato 3 ult quato 33. 4

25 5 33 [ ] [ ] l owv, 33 oly al t ca tat all t colou xal a wt t ag of t yot. c, uc ca t wll b o colou xal out of t ag of t yot, a o wll b qual to zo. Subttutg t valu to t quato ult 34. Itgato wt ct to wll ow ouc a fft quato, a t u lt o ow o log cotat wt ct to t tgato, but a vaabl t o. T colto of t tgato t ca t cto [ ] [ ] l T oly ot tuato wc t t a colou xal o t ot x aatg t ag of obl valu fo t tat of t yot a fo t, w t a o colou xal o t ot x wt t ag of t yot. I Fgu 5 abov, t a fou tuato wc coo to t ca. T a w t wol of ac yot btw x a yllow, btw yllow a yllow, btw yllow a yllow 3, o btw yllow 3 a x. W ca ft co a a xal t yot wt t at of t colou ac btw yllow a yllow 3. It ot obl to ly t t low lt o bot a to yllow, a t u lt o bot t a vaabl to yllow 3, a t woul allow ca wc t of t yot ca bfo t tat. Wat, to cota t ag of bl ot of yot, uc tat t ay oly occu oto btw t tat of t yot a yllow 3. T ca b acv by akg t low lt o qual to. Makg t cag to quato 3 ult quato [ ] [ ] l W ay ot, owv, tat uc ca, a t a o colou xal wt t ag of t yot, wll b qual to zo. c, t ca oly t ft tgato oul b clu. Makg t cag, a ttg qual to zo, ult quato 36. T tgato ov t ca t cto 5..5.

26 6 36 [ ] 00 T t ca fo fft ty of lt o t valu of cov all t obl lt o t tgato, o t ow obl to oc wt t tgato of quato 3, 34 a 35 wt ct to Itgatg ov w t lt of a cotat T cto colt t vato of a quato fo w t ag of locato fo t of t yot aat fo t ag of locato fo t tat by at lat o colou ot o t ot x, a c t lt o t tgato a all cotat. Ftly, t lt o t valu of quato 3 a ubttut to gv quato l l Rwtg t t tgato ult l l W oul ot tat t tgato of t t t wll av a cal ca w qual to -, a c t cay to aat t ca out of t ct uato. Makg t cag ult quato 39. Rwtg t quato t way au tat gat to o qual to t, o t wll b cay to co aot cal ca of t quato fo w qual to two.

27 7 39 l l fog t tgato ult [ ] [ ] l l 00 [ ] l l W t tgal a xa, by ubttutg t valu of t lt o t tgato fo, t ultg quato gv 4. T quato coct w t a at lat t colou xal wt t ag of t yot bg co, but w alo to v quato fo w t a oly two colou xal, w t oly a gl colou xal, o o colou xal at all, wt t ag of t yot bg co.

28 l l l l l l l l Ftly a quato wll b v fo t ca w t a o xal colou wt t ag of t yot, a c qual to zo. A ot abov, t quato wll cota oly a t coog to t ft tgal. T co tgal alcabl oly w gat ta o qual to o, t t w gat ta o qual to two, a t fout w gat ta o qual to t. W oly t ft tgal, a t cotat t ultl by, a clu t quato, a t valu of zo ubttut fo, t ult quato 4. T quato al all tac w t lt o tgato a cotat, a t ag of yot cota o colou xal. A t ag of tat a ot fo t yot ut b aat by cotut, t cotuty aft t tat but bfo t ut b t ot x, a t caot b a colou xal wt t ag Nxt a quato wll b v fo t ca tat t oly a gl colou xal wt t ag of t yot. A ot abov t quato ca b v by clug oly t ft two tgal. Takg t coog tgal fo quato 4, a ubttutg t valu o fo, ult quato 43. T quato al all ca w t lt o tgato a cotat, a t yot cota a gl colou xal l l l l Fally t cay to v o o quato, fo t ca w t yot clu xactly two colou xal wt t ag. w tak t ft t tgal of quato 4, a t qual to two. T ult quato 44, wc al all ca w t lt o tgato a cotat, but t yot cota two colou xal.

29 l l l l l l l l Fou fal quato av ow b v, quato 4, 4, 43 a 44. T yot cov all t ca wc t ag of tat valu a t ag of yot a aat by at lat o colou xal o t ot x, a wll b u t fal ltato of t ol Itgatg ov w t u lt o t o I t cto, quato a v fo tuato wc t yot u coato clu all of t colou xal a t ot x. Ft, ubttuto a a of t lt o quato 34, wc ult l l A wa t ca fo t tgato w t lt o w bot cotat, t tgato of t t t ct uato wll av a cal ca w qual to -, a o t ca ut b aat out of t uato o a to abl t tgato to b fo. Makg t cag ult quato l l W t tgato ov a fo t ult quato 47.

30 [ ] 00 l 00 l l Fally, ubttutg t lt o ouc quato 48. T t quato wc wll b u to calculat fo ag of t yot ac wc clu all of t colou xal a x, w t a at lat t colou xal. owv, w alo to co t ca w t a oly two colou xal, w t oly a gl colou xal, o w t a o colou xal at all l 00 00l00 00 l l Ft t ca w yot cota o colou xal, a c qual to zo wll b co. T quato wll cota oly t ft tgal, a w t valu of zo fo ubttut, t ult a gv owv, t ca tat quato 49 al, tat w t a o colou xal wt t ag of t yot, a c qual to zo, o colou xal ca av b obv at all, bcau t yot ut clu ay colou xal wc o xt. T a tat t

31 wll oly b a gl cotuty t yot ac, at t ot x. So, a w wll alway co t lagt obl ag of t yot ac tat obl wt ac u of a quato, t ca w wll co t ag of yot wc tat ayw t ag fo jut aft t ot x, gt ou to t ot of t a ot. c, t of t ag wll b 00 ut aft t tat, xbl wt t quato 00. Subttutg to 49 ug t quato ult quato Equato 50 tll u tat t u of t obablty of t ata gv a yot, ov t full ag of yot cotag ay gl ot, x, t a gal of w t colou ac tat ot, a qual to T quato ay to b faly agl, gv tat t al oly t ca tat t t ay ata, but t wll b uful calculatg t obablty tat atcula colou ay b wt t otato of a colou t t ca tat o colou xal av yt b obv. T co ca to co w t oly a gl colou xal. I t ca wll b qual to o, a t coog quato wll cota t coog to oly t ft two tgal quato 48. W t cag a a to t quato t ult a gv l00 00 l 00 Latly t a to v a quato fo w t a oly two colou xal, a c qual to two. T quato wll clu t coog to t ft t tgal of quato 48. Iclug jut t tgal a ttg qual to two ult quato l00 00 l 00 l 00 l T fou fal quato v t cto, 48, 50, 5 a 5 cov all ca wc t ag of t yot clu all t colou xal a x, a wll b u t fal cout ltato of t ol. 3

32 5..5 Itgatg ov w t low lt o t o T cto v a quato fo fo cotuou ag of t yot ac wc cota o colou xal o t ot x. Statg wt quato 36 v abov, ubttutg t valu fo t lt o ult quato Itgatg ov ouc quato A ubttutg t lt o ult quato 55. T quato wll b u to calculat t valu of fo all ag of yot wc cota t ay colou xal o t ot x Alyg t Equato T fal quato v cto 5. ca ow b u to calculat fo all go of t yot ac. T total of t ult of t calculato fo all t ag of t yot ac cotag x, a t total fo all t ag ot cotag x ca b calculat. T valu ca b ubttut to quato 6 to t t obablty tat t ot x co wt t otato of t colou t x. alculatg t obablty fo all obl valu of x wll gv t g of fuzzy b of ac u wt t otato of t colou t, tu fg a fuzzy t. 6 Lag t Dotato of Egl olou T fo Exal I o to vtgat ow t ol woul fo actc, a wat ot t la otato of t colou t woul av, t ol wa ta o t x coatc bac colou t of Egl wc a tgu cally o t ba of t u. Ug ata fo Bl a Kay 969, tat w a of t wt of t oologcal colou ac coog to ac colou t wt bg au alog t u o oly, a t tat w u to a ac colou t to a cto of t oologcal colou ac t ol, uc tat t x colou atto t colou ac. Bl a Kay ow t xtt of t colou o a aay of Mull colou c. A t Mull yt of colou c attt to ac t c oologcally v gaato of colou, t ub of colou c wt t ag of ac colou t oul at lat aoxat t z of t oologcal colou 3