Many Techniques Developed. Uncertainty. Aspects of Uncertainty. Decision Theory = Probability + Utility Theory
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1 Ucertait CSE 47 Ma Techiques Developed Fu Logic Certait Factors No-mootoic logic robabilit Ol oe has stood the test of time! UW CSE AI Facult Aspects of Ucertait Suppose ou have a flight at oo Whe should ou leave for SEATAC What are traffic coditios? How crowded is securit? Leavig 8 hours earl ma get ou there But? UW CSE AI Facult Decisio Theor = robabilit + Utilit Theor Mi before oo arrive-i-time 0 mi mi mi mi mi mi 0.99 Depeds o our prefereces Utilit theor: represetig & reasoig about prefereces UW CSE AI Facult 4
2 What Is robabilit? robabilit: Calculus for dealig with odetermiism ad ucertait Cf. Logic robabilistic model: Sas how ofte we epect differet thigs to occur Cf. Fuctio Wh Should You Care? The world is full of ucertait Logic is ot eough Computers eed to be able to hadle ucertait robabilit: ew foudatio for AI & CS! Massive amouts of data aroud toda Statistics ad CS are both about data Statistics lets us summarie ad uderstad it Statistics is the basis for most learig Statistics lets data do our work for us UW CSE AI Facult 5 UW CSE AI Facult 6 Outlie Logic vs. robabilit Basic otios Atomic evets probabilities joit distributio Iferece b eumeratio Idepedece & coditioal idepedece Baes rule Baesia etworks Statistical learig Damic Baesia etworks DBNs Markov decisio processes MDs Smbol: Q R Boolea values: T F State of the world: Assigmet to Q R Z Radom variable: Q Domai: ou specif e.g. {heads tails} [ 6] Atomic evet: complete specificatio of world: Q Z Mutuall eclusive Ehaustive rior probabilit aka Ucoditioal prob: Q Joit distributio: rob. of ever atomic evet UW CSE AI Facult 7 UW CSE AI Facult 8
3 True Sta for ropositios Aioms of robabilit Theor All probabilities betwee 0 ad 0 A true = false = 0. The probabilit of disjuctio is: A B A B A B A B A B UW CSE AI Facult 9 UW CSE AI Facult 0 rior robabilit Coditioal probabilit Coditioal or posterior probabilities e.g. cavit toothache = 0.8 i.e. give that toothache is all I kow Notatio for coditioal distributios: Cavit Toothache = -elemet vector of -elemet vectors If we kow more e.g. cavit is also give the we have cavit toothachecavit = New evidece ma be irrelevat allowig simplificatio: cavit toothache su = cavit toothache = 0.8 A questio ca be aswered b the joit distributio UW CSE AI Facult This kid of iferece sactioed b domai kowledge is crucial UW CSE AI Facult
4 True Coditioal robabilit Dilemma at the Detist s A B is the probabilit of A give B Assumes that B is the ol ifo kow. Defied b: A B A B B A A B B What is the probabilit of a cavit give a toothache? What is the probabilit of a cavit give the probe catches? UW CSE AI Facult CSE AI Facult 4 Iferece b Eumeratio Iferece b Eumeratio toothache= =.0 or 0% UW CSE AI Facult 5 toothache cavit =.0 +?? UW CSE AI Facult 6 4
5 Iferece b Eumeratio roblems?? Worst case time: Od Where d = ma arit Ad = umber of radom variables Space compleit also Od Sie of joit distributio How get Od etries for table?? UW CSE AI Facult 7 UW CSE AI Facult 8 Idepedece Idepedece A ad B are idepedet iff: A B A B A B These two costraits are logicall equivalet Therefore if A ad B are idepedet: A B A B A B A B A B UW CSE AI Facult 9 Complete idepedece is powerful but rare What to do if it does t hold? UW CSE AI Facult 0 5
6 Coditioal Idepedece Coditioal Idepedece II catch toothache cavit = catch cavit catch toothache cavit = catch cavit Wh ol 5 etries i table? Istead of 7 etries ol eed 5 UW CSE AI Facult UW CSE AI Facult ower of Cod. Idepedece Baes Formula Ofte usig coditioal idepedece reduces the storage compleit of the joit distributio from epoetial to liear!! Coditioal idepedece is the most basic & robust form of kowledge about ucertai eviromets. likelihood evidece prior UW CSE AI Facult UW CSE AI Facult 4 6
7 Use to Compute Diagostic robabilit from Causal robabilit Baes Rule & Cod. Idepedece E.g. let M be meigitis S be stiff eck M = S = 0. SM= 0.8 MS = UW CSE AI Facult 5 UW CSE AI Facult 6 Simple Eample of State Estimatio Suppose a robot obtais measuremet What is doorope? Causal vs. Diagostic Reasoig ope is diagostic. ope is causal. Ofte causal kowledge is easier to obtai. Baes rule allows us to use causal kowledge: cout frequecies! ope ope ope UW CSE AI Facult 7 UW CSE AI Facult 8 7
8 8 UW CSE AI Facult 9 Normaliatio au : au au : Algorithm: UW CSE AI Facult 0 Eample ope = 0.6 ope = 0. ope = ope = ope ope p ope ope p ope ope ope ope raises the probabilit that the door is ope. UW CSE AI Facult Combiig Evidece Suppose our robot obtais aother observatio. How ca we itegrate this ew iformatio? More geerall how ca we estimate...? UW CSE AI Facult Recursive Baesia Updatig Markov assumptio: is idepedet of... - if we kow i i
9 9 UW CSE AI Facult Eample: Secod Measuremet ope = 0.5 ope = 0.6 ope =/ ope ope ope ope ope ope ope lowers the probabilit that the door is ope.
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