Representing Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example

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1 C 188: Aificial Inelligence Fall 2007 epesening Knowledge ecue 17: ayes Nes III 10/25/2007 an Klein UC ekeley Popeies of Ns Independence? ayes nes: pecify complex join disibuions using simple local condiional disibuions Condiional independence makes his possible he gaph sucue of a N guaanees ceain condiional independences ecipe: shade evidence nodes Aemp 1: if wo nodes ae conneced by an undieced pah no blocked by a shaded node, hey ae condiionally independen Quesions: Wha independences does a N have? How o compue quaniies we wan fom quaniies we have Almos woks, bu no quie Whee does i beak? Answe: he v-sucue a doesn coun as a link in a pah unless shaded eachabiliy (he ayes all) Coec algoihm: hade in evidence a a souce node y o each age by seach aes: pai of (node, pevious sae ) uccesso funcion: unobseved: o any child o any paen if coming fom a child obseved: Fom paen o paen If you can each a node, i s condiionally independen of he sa node given evidence 1

2 Vaiables: : aining : affic : oof dips : I m sad Quesions: Causaliy? : affic When ayes nes eflec he ue causal paens: Ofen simple (nodes have fewe paens) Ofen easie o hink abou Ofen easie o elici fom expes asic affic ne e s muliply ou he join Ns need no acually be causal omeimes no causal ne exiss ove he domain E.g. conside he vaiables affic and ips End up wih aows ha eflec coelaion, no causaion Wha do he aows eally mean? opology may happen o encode causal sucue opology only guaaneed o encode condiional independencies 1/4 3/4 3/4 1/4 3/16 1/16 1/2 1/2 : evese affic evese causaliy? : Coins Exa acs don peven epesening independence, jus allow non-independence 9/16 7/16 1/3 2/3 1/7 6/7 3/16 1/ h h 1 2 h h h h h 2

3 Alenae Ns ummay ayes nes compacly encode join disibuions Guaaneed independencies of disibuions can be deduced fom N gaph sucue A ayes ne may have ohe independencies ha ae no deecable unil you inspec is specific disibuion he ayes ball algoihm (aka d-sepaaion) ells us when an obsevaion of one vaiable can change belief abou anohe vaiable Infeence eminde: Alam Newok Infeence: calculaing some saisic fom a join pobabiliy disibuion s: Poseio pobabiliy: Mos likely explanaion: Infeence by Enumeaion Given unlimied ime, infeence in Ns is easy ecipe: ae he maginal pobabiliies you need Figue ou A he aomic pobabiliies you need Calculae and combine hem : Whee did we use he N sucue? We didn! 3

4 Nomalizaion ick In his simple mehod, we only need he N o synhesize he join enies Nomalize Infeence by Enumeaion? Nesing ums Aomic infeence is exemely slow! lighly cleve way o save wok: Move he sums as fa igh as possible : Evaluaion ee View he nesed sums as a compuaion ee: Vaiable Eliminaion: Idea os of edundan wok in he compuaion ee We can save ime if we cache all paial esuls his is he basic idea behind vaiable eliminaion ill epeaed wok: calculae P(m a) P(j a) wice, ec. 4

5 asic Objecs ack objecs called facos Iniial facos ae local CPs uing eliminaion, ceae new facos Anaomy of a faco: 4 numbes, one fo each value of and E asic Opeaions Fis basic opeaion: join facos Combining wo facos: Jus like a daabase join uild a faco ove he union of he domains : Vaiables inoduced Vaiables summed ou Agumen vaiables, always nonevidence vaiables asic Opeaions econd basic opeaion: maginalizaion ake a faco and sum ou a vaiable hinks a faco o a smalle one A pojecion opeaion : Vaiable Eliminaion Wha you need o know: VE caches inemediae compuaions Polynomial ime fo ee-sucued gaphs! aves ime by maginalizing vaiables ask soon as possible ahe han a he end We will see special cases of VE lae You ll have o implemen he special cases Appoximaions Exac infeence is slow, especially when you have a lo of hidden nodes Appoximae mehods give you a (close) answe, fase 5