Today s class Probablty theory Bayesan nference From the ont dstrbuton Usng ndependence/factorng From sources of evdence Chapter 13 1 2 Sources of uncertanty Uncertan nputs Mssng data Nosy data Uncertan knowledge Multple causes lead to multple effects Incomplete enumeraton of condtons or effects Incomplete knowledge of causalty n the doman Probablstc/stochastc effects Uncertan outputs Abducton and nducton are nherently uncertan Default reasonng, even n deductve fashon, s uncertan Incomplete deductve nference may be uncertan Probablstc reasonng only gves probablstc results (summarzes uncertanty from varous sources Decson makng wth uncertanty Ratonal behavor: For each possble acton, dentfy the possble outcomes Compute the probablty of each outcome Compute the utlty of each outcome Compute the probablty-weghted (expected utlty over possble outcomes for each acton Select the acton wth the hghest expected utlty (prncple of Maxmum xpected Utlty 3 4 1
Why probabltes anyway? Kolmogorov showed that three smple axoms lead to the rules of probablty theory De Fnett, Cox, and Carnap have also provded compellng arguments for these axoms 1. All probabltes are between 0 and 1: 0 P(a 1 2. Vald propostons (tautologes have probablty 1, and unsatsfable propostons have probablty 0: P(true = 1 ; P(false = 0 3. The probablty of a dsuncton s gven by: P(a b = P(a + P(b P(a b a a b b 5 Random varables Doman Atomc event: complete specfcaton of state Pror probablty: degree of belef wthout any other evdence Jont probablty: matrx of combned probabltes of a set of varables Probablty theory Alarm, Burglary, arthquake Boolean (lke these, dscrete, contnuous Alarm=True Burglary=True arthquake=false alarm burglary earthquake P(Burglary =.1 P(Alarm, Burglary = alarm burglary.09.01 burglary.1.8 alarm 6 Probablty theory (cont. xample: Inference from the ont Condtonal probablty: probablty of effect gven causes Computng condtonal probs: P(a b = P(a b / P(b P(b: normalzng constant Product rule: P(a b = P(a b P(b Margnalzng: P(B = Σ a P(B, a P(B = Σ a P(B a P(a (condtonng P(burglary alarm =.47 P(alarm burglary =.9 P(burglary alarm = P(burglary alarm / P(alarm =.09 /.19 =.47 P(burglary alarm = P(burglary alarm P(alarm =.47 *.19 =.09 P(alarm = P(alarm burglary + P(alarm burglary =.09+.1 =.19 7 alarm alarm earthquake earthquake earthquake earthquake burglary.01.08.001.009 burglary.01.09.01.79 P(Burglary alarm = α P(Burglary, alarm = α [P(Burglary, alarm, earthquake + P(Burglary, alarm, earthquake = α [ (.01,.01 + (.08,.09 ] = α [ (.09,.1 ] Snce P(burglary alarm + P( burglary alarm = 1, α = 1/(.09+.1 = 5.26 (.e., P(alarm = 1/α =.19 quzlet: how can you verfy ths? P(burglary alarm =.09 * 5.26 =.474 P( burglary alarm =.1 * 5.26 =.526 8 2
xercse: Inference from the ont study prep Queres: What s the pror probablty of? What s the pror probablty of study? What s the condtonal probablty of prepared, gven study and? Save these answers for next tme! Independence When two sets of propostons do not affect each others probabltes, we call them ndependent, and can easly compute ther ont and condtonal probablty: Independent (A, B P(A B = P(A P(B, P(A B = P(A For example, {moon-phase, lght-level} mght be ndependent of {burglary, alarm, earthquake} Then agan, t mght not: Burglars mght be more lkely to burglarze houses when there s a new moon (and hence lttle lght But f we know the lght level, the moon phase doesn t affect whether we are burglarzed Once we re burglarzed, lght level doesn t affect whether the alarm goes off We need a more complex noton of ndependence, and methods for reasonng about these knds of relatonshps 9 10 xercse: Independence study prep Queres: Is ndependent of study? Is prepared ndependent of study? Condtonal ndependence Absolute ndependence: A and B are ndependent f P(A B = P(A P(B; equvalently, P(A = P(A B and P(B = P(B A A and B are condtonally ndependent gven C f P(A B C = P(A C P(B C Ths lets us decompose the ont dstrbuton: P(A B C = P(A C P(B C P(C Moon-Phase and Burglary are condtonally ndependent gven Lght-Level Condtonal ndependence s weaker than absolute ndependence, but stll useful n decomposng the full ont probablty dstrbuton 11 12 3
xercse: Condtonal ndependence study prep Queres: Is condtonally ndependent of prepared, gven study? Is study condtonally ndependent of prepared, gven? Bayes s rule Bayes s rule s derved from the product rule: P(Y X = P(X Y P(Y / P(X Often useful for dagnoss: If X are (observed effects and Y are (hdden causes, We may have a model for how causes lead to effects (P(X Y We may also have pror belefs (based on experence about the frequency of occurrence of effects (P(Y Whch allows us to reason abductvely from effects to causes (P(Y X. 13 14 Bayesan nference In the settng of dagnostc/evdental reasonng P H ( 1 m Know pror probablty of hypothess condtonal probablty Want to compute the posteror probablty Bayes s theorem (formula 1: H P( H P ( H = P( H P( H / P( hypotheses evdence/manfestatons P( H P( H P H ( Smple Bayesan dagnostc reasonng Knowledge base: vdence / manfestatons: 1, m Hypotheses / dsorders: H 1, H n and H are bnary; hypotheses are mutually exclusve (nonoverlappng and exhaustve (cover all possble cases Condtonal probabltes: P( H, = 1, n; = 1, m Cases (evdence for a partcular nstance: 1,, l Goal: Fnd the hypothess H wth the hghest posteror Max P(H 1,, l 15 16 4
Bayesan dagnostc reasonng II Bayes rule says that P(H 1,, l = P( 1,, l H P(H / P( 1,, l Assume each pece of evdence s condtonally ndependent of the others, gven a hypothess H, then: P( 1,, l H = l =1 P( H If we only care about relatve probabltes for the H, then we have: P(H 1,, l = α P(H l =1 P( H Lmtatons of smple Bayesan nference Cannot easly handle mult-fault stuatons, nor cases where ntermedate (hdden causes exst: Dsease D causes syndrome S, whch causes correlated manfestatons M 1 and M 2 Consder a composte hypothess H 1 H 2, where H 1 and H 2 are ndependent. What s the relatve posteror? P(H 1 H 2 1,, l = α P( 1,, l H 1 H 2 P(H 1 H 2 = α P( 1,, l H 1 H 2 P(H 1 P(H 2 = α l =1 P( H 1 H 2 P(H 1 P(H 2 How do we compute P( H 1 H 2?? 17 18 Lmtatons of smple Bayesan nference II Assume H1 and H2 are ndependent, gven 1,, l? P(H 1 H 2 1,, l = P(H 1 1,, l P(H 2 1,, l Ths s a very unreasonable assumpton arthquake and Burglar are ndependent, but not gven Alarm: P(burglar alarm, earthquake << P(burglar alarm Another lmtaton s that smple applcaton of Bayes s rule doesn t allow us to handle causal channg: A: ths year s weather; B: cotton producton; C: next year s cotton prce A nfluences C ndrectly: A B C P(C B, A = P(C B Need a rcher representaton to model nteractng hypotheses, condtonal ndependence, and causal channg Next tme: condtonal ndependence and Bayesan networks! 19 5