Modeling uncertainty using probabilities

Transcription

1 S 1571 Itroduto to I Leture 23 Modelg uertty usg probbltes Mlos Huskreht mlos@s.ptt.edu 5329 Seott Squre dmstrto Fl exm: Deember :00-1:50pm 5129 Seott Squre

2 Uertty To mke dgost feree possble we eed to represet kowledge xoms tht relte symptoms d dgoss eumo leess Fever ough WB out roblem: dsese/symptoms reltos re ot determst They re uert or stohst d vry from ptet to ptet Modelg the uertty. ey hlleges: How to represet the reltos the presee of uertty? How to mpulte suh kowledge to mke ferees? Hums reso wth uertty. eumo? leess Fever ough WB out

3 Methods for represetg uertty robblty theory well defed theory for modelg d resog the presee of uertty turl hoe to reple ertty ftors Fts propostol sttemets re represeted v rdom vrbles wth two or more vlues Exmple: eumo s rdom vrble vlues: True d Flse Eh vlue be heved wth some probblty: eumo True WBout hgh Modelg uertty wth probbltes robblst exteso of propostol log. ropostos: sttemets bout the world Represeted by the ssgmet of vlues to rdom vrbles Rdom vrbles:! Boole eumo s ether True Flse Rdom vrble Vlues! Mult-vlued s oeof { Nop Mld Moderte Severe} Rdom vrble Vlues otuous HertRte s vlue < 0 ; 250 > Rdom vrble Vlues

4 robbltes Uodtol probbltes pror probbltes eumo or eumo Flse WBout hgh eumo True robblty dstrbuto Defes probbltes for ll possble vlue ssgmets to rdom vrble Vlues re mutully exlusve eumo True eumo Flse eumo True Flse eumo robblty dstrbuto Defes probblty for ll possble vlue ssgmets Exmple 1: eumo True eumo Flse eumo True Flse eumo True + eumo Flse 1 robbltes sum to 1!!! eumo Exmple 2: WBout hgh WBout orml WBout hgh WBout WBout hgh orml low

5 Jot probblty dstrbuto Jot probblty dstrbuto for set vrbles Defes probbltes for ll possble ssgmets of vlues to vrbles the set Exmple: vrbles eumo d WBout peumo WBout Is represeted by 2 3 mtrx eumo True Flse WBout hgh orml low Jot probbltes Mrglzto redues the dmeso of the jot dstrbuto Sums vrbles out peumo WBout eumo True Flse 2 3 mtrx WBout hgh orml low eumo WBout Mrglzto here summg of olums or rows

6 Full jot dstrbuto the jot dstrbuto for ll vrbles the problem It defes the omplete probblty model for the problem Exmple: peumo dgoss Vrbles: eumo Fever leess WBout ough Full jot defes the probblty for ll possble ssgmets of vlues to eumo Fever leess WBout ough eumo T WBout Hgh Fever T ough T leess T eumo T WBout Hgh Fever T ough T leess F eumo T WBout Hgh Fever T ough F leess T et odtol probbltes odtol probblty dstrbuto Defes probbltes for ll possble ssgmets gve fxed ssgmet to some other vrble vlues eumo true WBout hgh eumo WBout 3 elemet vetor of 2 elemets WBout hgh orml low eumo True Flse eumo true WBout hgh + eumo flse WBout hgh

7 odtol probbltes odtol probblty Is defed terms of the jot probblty: B B s.t. B 0 B Exmple: peumo true WBout hgh peumo true WBout hgh WBout hgh peumo flse WBout hgh peumo flse WBout hgh WBout hgh odtol probbltes odtol probblty dstrbuto. B B s.t. B 0 B rodut rule. Jo probblty be expressed terms of odtol probbltes B B B h rule. y jot probblty be expressed s produt of odtols

8 Byes rule odtol probblty. B B B B B Byes rule: B B B Whe s t useful? Whe we re terested omputg the dgost query from the usl probblty effet use use use effet effet Reso: It s ofte eser to ssess usl probblty E.g. robblty of peumo usg fever vs. probblty of peumo gve fever Byes Rule smple dgost feree. Deve equpmet opertg ormlly or mlfutog. Operto of the deve sesed dretly v sesor Sesor redg s ether hgh or low Deve sttus Deve sttus orml mlfutog Sesor redg Deve sttus Sesor redg Deve\Sesor hgh low orml mlfutog

9 Byes Rule smple dgost feree. Dgost feree: ompute the probblty of deve opertg ormlly or mlfutog gve sesor redg Deve sttus Sesor redg hgh? Deve sttus orml Deve sttus mlfuto g Sesor redg Sesor redg hgh hgh Note tht typlly the opposte odtol probbltes re gve to us: they re muh eser to estmte Soluto: pply Byes rule to reverse the odtog vrbles Vrous feree tsks: robblst feree Dgost tsk. from effet to use eumo Fever T redto tsk. from use to effet Fever eumo T Other probblst queres queres o jot dstrbutos. Fever Fever hest

10 Iferee y query be omputed from the full jot dstrbuto!!! Jot over subset of vrbles s obted through mrglzto odtol probblty over set of vrbles gve other vrbles vlues s obted through mrglzto d defto of odtols j d j D b B j j d D b B d D b B d D d D Iferee y query be omputed from the full jot dstrbuto!!! y jot probblty be expressed s produt of odtols v the h rule. Sometmes t s eser to defe the dstrbuto terms of odtol probbltes: E.g T eumo Fever F eumo Fever

11 Modelg uertty wth probbltes Defg the full jot dstrbuto mkes t possble to represet d reso wth uertty uform wy We re ble to hdle rbtrry feree problem roblems: Spe omplexty. To store full jot dstrbuto we eed to remember Od umbers. umber of rdom vrbles d umber of vlues Iferee tme omplexty. To ompute some queres requres Od. steps. qusto problem. Who s gog to defe ll of the probblty etres? Medl dgoss exmple Spe omplexty. eumo 2 vlues: TF Fever 2: TF ough 2: TF WBout 3: hgh orml low pleess 2: TF Number of ssgmets: 2*2*2*3*248 We eed to defe t lest 47 probbltes. Tme omplexty. ssume we eed to ompute the mrgl of eumot from the full jot eumo T Fever ough T F j T F k h l u T F Sum over: 2*2*3*224 ombtos j WBout k le u

12 Modelg uertty wth probbltes owledge bsed system er 70s erly 80 s Extesol o-probblst models Solve the spe tme d qusto bottleeks probblty-bsed models froze the developmet d dvemet of B systems d otrbuted to the slow-dow of I 80s geerl Brekthrough lte 80s begg of 90s Byes belef etworks Gve solutos to the spe qusto bottleeks rtl solutos for tme omplextes Byes belef etwork Byes belef etworks BBNs Byes belef etworks. Represet the full jot dstrbuto over the vrbles more omptly wth smller umber of prmeters. Tke dvtge of odtol d mrgl depedees mog rdom vrbles d B re depedet B B d B re odtolly depedet gve B B B