Bayesian belief networks
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- Evelyn Davis
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1 Lecture 19 ayesa belef etworks los Hauskrecht 539 Seott Square Varous ferece tasks: robablstc ferece Dagostc task. from effect to cause eumoa Fever redcto task. from cause to effect Fever eumoa Other probablstc queres queres o jot dstrbutos. Fever Fever Chesta
2 Iferece y query ca be computed from the full jot dstrbuto!!! ot over a subset of varables s obtaed through margalzato Codtoal probablty over set of varables gve other varables values s obtaed through margalzato ad defto of codtoals j d j D c C b a c C a j j d D c C b a d D c C b a c C a d D c C a c C a d D Iferece y query ca be computed from the full jot dstrbuto!!! y jot probablty ca be expressed as a product of codtoals va the cha rule. Sometmes t s easer to defe the dstrbuto terms of codtoal probabltes:.g K K K K K K K eumoa Fever F eumoa Fever
3 odelg ucertaty wth probabltes Defg the full jot dstrbuto makes t possble to represet ad reaso wth ucertaty a uform way We are able to hadle a arbtrary ferece problem roblems: Space complexty. o store a full jot dstrbuto we eed to remember Od umbers. umber of radom varables d umber of values Iferece tme complexty. o compute some queres requres Od. steps. cqusto problem. Who s gog to defe all of the probablty etres? edcal dagoss example Space complexty. eumoa values: F Fever : F Cough : F WCcout 3: hgh ormal low paleess : F Number of assgmets: ***3*48 We eed to defe at least 47 probabltes. me complexty. ssume we eed to compute the margal of eumoa from the full jot eumoa F j F k h l u F eumoa Fever Cough j WCcout k ale u Sum over: **3*4 combatos
4 odelg ucertaty wth probabltes Kowledge based system era 70s early 80 s xtesoal o-probablstc models Solve the space tme ad acqusto bottleecks probablty-based models froze the developmet ad advacemet of K systems ad cotrbuted to the slow-dow of I 80s geeral reakthrough late 80s begg of 90s ayesa belef etworks Gve solutos to the space acqusto bottleecks artal solutos for tme complextes ayesa belef etwork ayesa belef etworks Ns ayesa belef etworks. Represet the full jot dstrbuto over the varables more compactly wth a smaller umber of parameters. ake advatage of codtoal ad margal depedeces amog radom varables ad are depedet ad are codtoally depedet gve C C C C C C
5 larm system example. ssume your house has a alarm system agast burglary. You lve the sesmcally actve area ad the alarm system ca get occasoally set off by a earthquake. You have two eghbors ary ad oh who do ot kow each other. If they hear the alarm they call you but ths s ot guarateed. We wat to represet the probablty dstrbuto of evets: urglary arthquake larm ary calls ad oh calls Causal relatos urglary arthquake larm arycalls ayesa belef etwork. 1. Drected acyclc graph Nodes radom varables urglary arthquake larm ary calls ad oh calls Lks drect causal depedeces betwee varables. he chace of larm s flueced by arthquake he chace of oh callg s affected by the larm urglary arthquake larm arycalls
6 ayesa belef etwork.. Local codtoal dstrbutos relate varables ad ther parets urglary arthquake larm arycalls ayesa belef etwork. urglary F F larm F F arthquake F F F F F arycalls F F
7 ayesa belef etworks geeral wo compoets: S ΘS Drected acyclc graph Nodes correspod to radom varables ssg lks ecode depedeces arameters Local codtoal probablty dstrbutos for every varable-paret cofgurato pa Where: pa - stad for parets of F F F F F Full jot dstrbuto Ns Full jot dstrbuto s defed terms of local codtoal dstrbutos obtaed va the cha rule: 1.. xample: 1.. ssume the followg assgmet of values to radom varables F pa he ts probablty s: F F
8 ayesa belef etworks Ns ayesa belef etworks Represet the full jot dstrbuto over the varables more compactly usg the product of local codtoals. ut how dd we get to local parameterzatos? swer: Graphcal structure ecodes codtoal ad margal depedeces amog radom varables ad are depedet ad are codtoally depedet gve C C C C C C he graph structure mples the decomposto!!! Idepedeces Ns 3 basc depedece structures: urglary urglary arthquake larm larm larm arycalls
9 Idepedeces Ns urglary urglary arthquake larm larm larm arycalls 1. s depedet of urglary gve larm Idepedeces Ns urglary urglary arthquake larm larm larm arycalls. urglary s depedet of arthquake ot kowg larm urglary ad arthquake become depedet gve larm!!
10 Idepedeces Ns 1.. urglary urglary arthquake 3. larm larm larm arycalls 3. arycalls s depedet of gve larm Idepedeces N N dstrbuto models may codtoal depedece relatos amog dstat varables ad sets of varables hese are defed terms of the graphcal crtero called d- separato D-separato ad depedece Let Y ad Z be three sets of odes If ad Y are d-separated by Z the ad Y are codtoally depedet gve Z D-separato : s d-separated from gve C f every udrected path betwee them s blocked wth C ath blockg 3 cases that expad o three basc depedece structures
11 Udrected path blockg s d-separated from gve C f every udrected path betwee them s blocked 1. ath blockg wth a lear substructure Z Y Z C Y Udrected path blockg s d-separated from gve C f every udrected path betwee them s blocked. ath blockg wth the wedge substructure Z Z C Y Y
12 Udrected path blockg s d-separated from gve C f every udrected path betwee them s blocked 3. ath blockg wth the vee substructure Z Y Y Z or ay of ts descedats ot C Idepedeces Ns urglary arthquake larm RadoReport arycalls arthquake ad urglary are depedet gve arycalls?
13 Idepedeces Ns urglary arthquake larm RadoReport arycalls arthquake ad urglary are depedet gve arycalls F urglary ad arycalls are depedet ot kowg larm? Idepedeces Ns urglary arthquake larm RadoReport arycalls arthquake ad urglary are depedet gve arycalls F urglary ad arycalls are depedet ot kowg larm F urglary ad RadoReport are depedet gve arthquake?
14 Idepedeces Ns urglary arthquake larm RadoReport arycalls arthquake ad urglary are depedet gve arycalls F urglary ad arycalls are depedet ot kowg larm F urglary ad RadoReport are depedet gve arthquake urglary ad RadoReport are depedet gve arycalls? Idepedeces Ns urglary arthquake larm RadoReport arycalls arthquake ad urglary are depedet gve arycalls F urglary ad arycalls are depedet ot kowg larm F urglary ad RadoReport are depedet gve arthquake urglary ad RadoReport are depedet gve arycalls F
15 ayesa belef etworks Ns ayesa belef etworks Represets the full jot dstrbuto over the varables more compactly usg the product of local codtoals. So how dd we get to local parameterzatos? pa he decomposto s mpled by the set of depedeces ecoded the belef etwork. Full jot dstrbuto Ns Rewrte the full jot probablty usg the product rule: F
16 Full jot dstrbuto Ns F F F F Rewrte the full jot probablty usg the product rule: Full jot dstrbuto Ns F F F F F F Rewrte the full jot probablty usg the product rule:
17 Full jot dstrbuto Ns F F F F F F Rewrte the full jot probablty usg the product rule: Full jot dstrbuto Ns F F F F F F Rewrte the full jot probablty usg the product rule:
18 Full jot dstrbuto Ns Rewrte the full jot probablty usg the product rule: F F F F F F F ayesa belef etwork. 1. Drected acyclc graph Nodes radom varables Lks mssg lks ecode depedeces. urglary arthquake larm arycalls
19 ayesa belef etwork. Local codtoal dstrbutos relate varables ad ther parets F F urglary arthquake larm F F F F F F F arycalls F F Full jot dstrbuto Ns Full jot dstrbuto s defed terms of local codtoal dstrbutos obtaed va the cha rule: 1.. xample: 1.. ssume the followg assgmet of values to radom varables F pa he ts probablty s: F F
20 arameter complexty problem I the N the full jot dstrbuto s defed as: 1.. pa 1.. What dd we save? larm example: 5 bary rue False varables # of parameters of the full jot: urglary 5 3 Oe parameter s for free: # of parameters of the N:? larm arthquake arycalls arameter complexty problem I the N the full jot dstrbuto s defed as: 1.. pa 1.. What dd we save? larm example: 5 bary rue False varables # of parameters of the full jot: urglary 5 3 Oe parameter s for free: # of parameters of the N: Oe parameter every codtoal s for free:? larm arthquake arycalls
21 arameter complexty problem I the N the full jot dstrbuto s defed as: 1.. pa 1.. What dd we save? larm example: 5 bary rue False varables # of parameters of the full jot: urglary 5 3 Oe parameter s for free: # of parameters of the N: Oe parameter every codtoal s for free: larm arthquake arycalls
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