CIS587 - Artificial Intelligence. Uncertainty CIS587 - AI. KB for medical diagnosis. Example.

Transcription

1 CIS587 - rtfcl Intellgence Uncertnty K for medcl dgnoss. Exmple. We wnt to uld K system for the dgnoss of pneumon. rolem descrpton: Dsese: pneumon tent symptoms fndngs, l tests: Fever, Cough, leness, WC whte lood cells count, Chest pn, etc. Representton of ptent cse: Sttements tht hold re true for tht ptent. E.g: Fever True Cough Flse WCcountHgh Dgnostc tsk: we wnt to nfer whether the ptent suffers from the pneumon or not gven the symptoms 1

2 Uncertnty To mke dgnostc nference possle we need to represent rules or xoms tht relte symptoms nd dgnoss rolem: dsese/symptoms relton s not determnstc thngs my vry from ptent to ptent t s uncertn Dsese Symptoms uncertnty ptent sufferng from pneumon my not hve fever ll the tmes, my or my not hve cough, whte lood cell test cn e n norml rnge. Symptoms Dsese uncertnty Hgh fever s typcl for mny dseses e.g. cterl dseses nd does not pont specfclly to pneumon Fever, cough, pleness, hgh WC count comned do not lwys pont to pneumon Modelng the uncertnty. Relton etween the dsese nd symptoms s not determnstc. Key ssues: How to descre the reltons n the presence of uncertnty? How to mnpulte such knowledge to mke nferences? Humns cn reson wth uncertnty.? 2

3 Uncertnty Reltons t the level of detl we consder re not determnstc, they re uncertn Resons for uncertnty nd the need to hndle t: Effcency, cpcty lmts It s often mpossle to enumerte nd model ll components of the world nd ther reltons Oservlty It s mpossle to oserve ll relevnt components of the world. Humns cn reson wth uncertnty!!! Cn computer systems do the sme? We need formlsms to model nd mnpulte uncertnty. Methods for representng uncertnty Defult or non-monotonc logc Sttements uld on ssumptons tht cn e retrcted. Exmples: ssume tht the cr does not hve flt tre ssume tht cr component works unless there s n evdence of the contrry. Sttements consdered to e true, unless new nformton gnst them s presented. Sttements re retrcted or overrdden rolem: excepton hndlng, the need to enumerte ll exceptons n whch ssumptons do not hold 3

4 Methods for representng uncertnty Extend formlsms sed on propostonl nd frst-order logc to reflect uncertn, mprecse sttements reltons Typclly rules wth vrous fudge fctors opulr n 70-80s n knowledge-sed systems e.g.,mycin If Then 1. The stn of the orgnsm s grm-postve, nd 2. The morphology of the orgnsm s coccus, nd 3. The growth conformton of the orgnsm s chns wth certnty 0.7 the dentty of the orgnsm s streptococcus rolems: Chnng of multple nference rules propgton of uncertnty Comntons of rules wth the sme conclusons fter some numer of comntons results not ntutve Representng uncertnty wth certnty fctors Fcts propostonl sttements re ssgned some certnty numer reflectng the elef n tht the sttement s stsfed: CF neumon True 0.7 Rules ncorporte tests on the certnty vlues n [0.5,1] n [0.7,1] C wth CF 0.8 Comnton of multple rules n [0.5,1] n [0.7,1] C wth CF 0.8 E n [0.8,1] D n [0.9,1] C wth CF 0.9 CF C mx[0.9;0.8] 0.9 CF C 0.9* CF C * ? 4

5 Methods for representng uncertnty rolty theory roposton sttements represented y rndom vrles nd the ssgnment of two or more vlues to vrles Ech vlue cn e cheved wth some prolty: neumon True WCcount hgh Cn model the effect of fndngs: neumon True Fever True 0.02 neumon True Fever True, WCcount hgh, Cough True 0.4 Suectve or yesn prolty: roltes relte propostons to one own stte of knowledge, nd not ssertons out the world. rolty theory Well-defned theory for representng nd mnpultng sttements wth uncertnty xoms of prolty: For ny two propostons, True 1 nd Flse

6 Modelng uncertnty wth proltes ssume the extenson of propostonl logc. ropostons: sttements out the world ssgnment of vlues to rndom vrles Rndom vrles: oolen neumon s ether True, Flse Mult-vlued WCcount s ether Hgh, Norml, Low roltes Uncondtonl proltes pror proltes neumon or neumon True WCcount hgh rolty dstruton Defnes prolty vlues for ll possle ssgnments neumon True neumon Flse roltes sum to 1!!! neumon True Flse neumon True + neumon Flse 1 neumon

7 rolty dstruton rolty dstruton Defnes prolty vlues for ll possle ssgnments WCcount hgh WCcount WCcount WCcount norml hgh norml WCcount hgh low Jont prolty dstruton for set of vrles Defnes proltes for ll possle ssgnments to vlues of vrles n the set WCcount pneumon, WCcount hgh norml low neumon True Flse Jont proltes Jont prolty dstruton for set of vrles Defnes proltes for ll possle ssgnments to vlues of vrles n the set pneumon, WCcount 2 3 mtrx neumon WCcount hgh norml low neumon True Flse WCcount Mrgnlzton summng of rows, or columns - summng out vrles

8 Condtonl proltes Condtonl prolty dstruton Defnes proltes for ll possle ssgnments, gven fxed ssgnment for some other vrle vlues neumon true WCcount hgh pneumon WCcount 3 element vector of 2 elements WCcount hgh norml low neumon True Flse neumon true WCcount hgh + neumon flse WCcount hgh Condtonl proltes Condtonl prolty dstruton. Defned n terms of ont prolty, s.t. 0 pneumon true, WCcount hgh pneumon true WCcount hgh WCcount hgh roduct rule. Jon prolty cn e expressed n terms of condtonl proltes, Chn rule. ny ont cn e expressed s product of condtonls X1, X 2, X n X n X1, X n 1 X1, X n 1 X n X1, X n 1 X n 1 X1, X n 2 X1, X n 2 n X X X 1 1, 1 8

9 9 yes rule Condtonl prolty. yes rule: When s t useful? When nterested n computng the dgnostc prolty, from the cusl prolty Reson: It s often eser to ssess cusl prolty E.g. rolty of pneumon cusng fever vs. prolty of pneumon gven fever,, effect cuse cuse effect effect cuse yes rule ssume vrle wth multple vlues: yes rule cn e rewrtten s: k 1, 2, 1 k β 1/ 1 k β for ll vlues of k 1, 2, 1. compute for ll, nd 2. otn the result y renormlzng the prolty vector wth Used n prctce when we wnt to compute: β

10 Full ont dstruton the ont dstruton for ll vrles n the prolem, full ont prolty dstruton, defnes the complete prolty model Exmple: pneumon dgnoss Full ont defnes the prolty for ll possle ssgnments of vlues to neumon, Fever, leness, WCcount, Cough neumon T, WCcount Hgh, Fever T, Cough T, leness T neumon T, WCcount Hgh, Fever T, Cough T, leness F neumon T, WCcount Hgh, Fever T, Cough F, leness T ny prolstc query cn e otned computed from the full ont prolty etc Full ont dstruton Computton of prolstc nference queres Jont over smller numer of vrles s otned through mrgnlzton, C c,, C c, D Condtonl prolty over set of vrles, gven other vrles vlues s otned through mrgnlzton nd defnton of condtonls, C c, D d D d, C c, C c d,, C c, D d,, C c, D d 10

11 Modelng uncertnty wth proltes Defnng the full ont dstruton mkes t possle to represent nd reson wth uncertnty n unform wy We re le to hndle n rtrry nference prolem rolems: Spce complexty. To store full ont dstruton we need to rememer Od n numers. n:numer of rndom vrles, d :numer of vlues Inference tme complexty. To compute some queres requres Od. n steps. cquston prolem. Who s gong to defne ll of the prolty entres? Medcl dgnoss exmple Spce complexty neumon 2 vlues: T,F, Fever 2: T,F, Cough 2: T,F, WCcount 3: hgh, norml, low, pleness 2: T,F Numer of ssgnments: 2*2*2*3*248 We need to defne t lest 47 proltes. Tme complexty ssume we need to compute the mrgnl of neumont from the full ont neumon T Fever, Cough T, F T, F k h, n, l u T, F Sum over: 2*2*3*224 comntons, WCcount k, le u 11

12 Modelng uncertnty wth proltes Knowledge sed system er 70s erly 80 s Extensonl non-prolstc models Spce, tme nd cquston ottlenecks n proltysed models froze the development nd dvncement of K systems nd contruted to the slow-down of I n 80s n generl rekthrough lte 80s, egnnng of 90s yesn elef networks Gve solutons to the spce, cquston ottlenecks rtl solutons for tme complextes 12