Uncertain t y Chapter 14 c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 1
Outline } Uncertaint y } Probabili t y } Syntax } Semantics } Inference rules c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 2
Uncertain t y Let action A = leave fo r airp o rt t minutes b efo re �ight t Will A get me there on time? t Problems: 1) pa rtial observabili t y (road state, other drivers' plans, etc.) 2) noisy senso rs (K CBS tra�c rep o rts) 3) uncertain t y in action outcomes (�at tire, etc.) 4) immense complexit y of mo delling and p redicting tra�c Hence a purely logical app roach either 1) risks falseho o d: \ A will get me there on time" 25 o r 2) leads to conclusions that a re to o w eak fo r decision making: \ A will get me there on time if there's no accident on the b ridge 25 and it do esn't rain and my tires remain intact etc etc." ( A might reasonably b e said to get me there on time 1440 : : : ) but I'd have to sta y overnight in the airp o rt c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 3
Metho ds for handling uncertain t y Default o r nonmonotonic logic: Assume my ca r do es not have a �at tire Assume A w o rks unless contradicted b y evidence 25 Issues: What assumptions a re reasonable? Ho w to handle contradicti on? Rules with fudge facto rs : A 7! get there on time 25 0 : 3 S pr ink l er 7! W etGr ass 0 : 99 W etGr ass 7! R ain 0 : 7 S pr ink l er R ain ?? Issues: Problems with combinati on, e.g., causes Probabili t y Given the available evidence, A will get me there on time with p robabili t y 0.04 25 Mahaviraca ry a (9th C.), Ca rdamo (1565) theo ry of gambling (F uzzy logic handles de gr e e of truth NOT uncertaint y e.g., W etGr ass is true to degree 0.2) c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 4
Probabilit y Probabili stic assertions summarize e�ects of laziness : failure to enumerate exceptions, quali�cati ons, etc. igno rance : lack of relevant facts, initial conditions, etc. Subjective o r Ba y esian p robabili t y: Probabili ties relate p rop ositions to one's o wn state of kno wledge ( A ) = 0 : 06 e.g., P j no rep o rted accidents 25 These a re not assertions ab out the w o rld Probabili ties of p rop ositions change with new evidence: ( A a.m. ) = 0 : 15 e.g., P j no rep o rted accidents ; 5 25 = (Analogous to logical entailment status K B j � , not truth.) c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 5
Making decisions under uncertain t y Supp ose I b elieve the follo wing: ( A : ) = 0 : 04 P gets me there on time j : : 25 ( A j : ) = 0 : 70 P gets me there on time : : 90 P ( A j : : : ) = 0 : 95 gets me there on time 120 ( A : ) = 0 : 9999 P j : : gets me there on time 1440 Which action to cho ose? Dep ends on my p references fo r missing �ight vs. airp o rt cuisine, etc. Utilit y theo ry is used to rep resent and infer p references Decision theo ry = utilit y theo ry + p robabili t y theo ry c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 6
Axioms of probabilit y F o r any p rop ositions A , B 0 ( A ) 1 1. � P � ( T ue ) = 1 ( F se ) = 0 2. P r and P al ( A ) = ( A ) + ( B ) ( A ) 3. P _ B P P � P ^ B True A B A B > de Finetti (1931): an agent who b ets acco rding to p robabiliti es that violate these axioms can b e fo rced to b et so as to lose money rega rdless of outcome. c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 7
Syn tax Simila r to p rop ositional logic: p ossible w o rlds de�ned b y assignment of values to random va riables . Prop ositiona l o r Bo olean random va riables e.g., C av ity (do I have a cavit y?) Include p rop ositiona l logic exp ressions e.g., : B ur g l ar y _ E ar thq uak e Multivalued random va riables e.g., W eather is one of h sunny ; r ain; cl oudy ; snow i V alues must b e exhaustive and mutually exclusive Prop osition constructed b y assignment of a value: = = e.g., W eather sunny ; also C av ity tr ue fo r cla rit y c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 8
Syn tax con td. Prio r o r uncondition al p robabili tie s of p rop osition s ( C ) = 0 : 1 ( W = ) = 0 : 72 e.g., P av ity and P eather sunny co rresp ond to b elief p rio r to a rrival of any (new) evidence Probabili t y distribution gives values fo r all p ossible assignments: P ( W ) = h 0 : 72 ; 0 : 1 ; 0 : 08 ; 0 : 1 i eather (no rmalize d , i.e., sums to 1) Joint p robabilit y distributi on fo r a set of va riables gives values fo r each p ossible assignment to all the va riables P ( W ) 4 2 eather ; C av ity = a � matrix of values: = W eather sunny r ain cl oudy snow = C av ity tr ue = C av ity f al se c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 9
Syn tax con td. Conditional o r p osterio r p robabili tie s ( C oothache ) = 0 : 8 e.g., P av ity j T i.e., given that T oothache is all I kno w Notation fo r conditiona l distributi ons: P ( W e ) eather j E ar thq uak = 2-element vecto r of 4-element vecto rs If w e kno w mo re, e.g., C av ity is also given, then w e have ( C ) = 1 P av ity j T oothache; C av ity Note: the less sp eci�c b elief r emains valid after mo re evidence a rrives, but is not alw a ys useful New evidence ma y b e irrelevant , allo wing simpli�cati on, e.g., ( C 49 er in ) = ( C oothache ) = 0 : 8 P av ity j T oothache; sW P av ity j T This kind of inference, sanctioned b y domain kno wledge, is crucial c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 10
Conditional probabilit y De�nition of conditional p robabili t y: ( A ^ ) P B ( A j B ) = ( B ) 6 = 0 P if P ( B ) P Pro duct rule gives an alternative fo rmulation: ( A ) = ( A j B ) P ( B ) = ( B j A ) P ( A ) P ^ B P P A general version holds fo r whole distribution s, e.g., P ( W ) = P ( W j C ) P ( C ) eather ; C av ity eather av ity av ity 4 � 2 (View as a set of equations, not matrix mult.) Chain rule is derived b y successive applicati on of p ro duct rule: P ( X ) = P ( X ) P ( X j X ) ; : : : ; X ; : : : ; X ; : : : ; X 1 1 n � 1 1 n � 1 n n ( X ) P ( X j X ) ( X j X ) = P ; : : : ; X ; : : : ; X P ; : : : ; X 1 n � 2 1 n � 2 1 n � 1 n n 1 : : : = n � P ( X ) j X ; : : : ; X = 1 i � 1 i = 1 i c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 11
Ba y es' Rule ( A ) = ( A j B ) P ( B ) = ( B j A ) P ( A ) Pro duct rule P ^ B P P ( B j A ) P ( A ) P ( A j B ) = ) Ba y es' rule P ( B ) P Why is this useful??? F o r assessing diagnostic p robabili t y from causal p robabilit y: ( E ause ) P ( e ) P f f ect j C C aus ( C ect ) = P ause j E f f ( E ect ) P f f E.g., let M b e meningitis, S b e sti� neck: ( S j M ) P ( M ) 0 : 8 � 0 : 0001 P ( M j S ) = = = 0 : 0008 P ( S ) 0 : 1 P Note: p osterio r p robabilit y of meningiti s still very small! c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 12
Normalization Supp ose w e wish to compute a p osterio r distributi on over A = given B b , and supp ose A has p ossible values a : : : a 1 m W e can apply Ba y es' rule fo r each value of A : ( A = j B = b ) = ( B = b j A = ) P ( A = ) =P ( B = b ) P a P a a 1 1 1 : : : ( A = = b ) = ( B = = ) P ( A = ) =P ( B = b ) P a j B P b j A a a m m m � ( A = = b ) = 1 : P a j B Adding these up, and noting that i i 1 = � 1 =P ( B = b ) = ( B = b j A = ) P ( A = ) P a a i i i This is the no rmalizati on facto r , constant w.r.t. i , denoted � : ( A j B = b ) = � P ( B = b j A ) P ( A ) P T ypically compute an unno rmalize d distributi on, no rmalize at end ( B = b j A ) P ( A ) = h 0 : 4 ; 0 : 2 ; 0 : 2 i e.g., supp ose P h 0 : 4 ; 0 : 2 ; 0 : 2 i P ( A j B = b ) = � h 0 : 4 ; 0 : 2 ; 0 : 2 i = = h 0 : 5 ; 0 : 25 ; 0 : 25 i then 0 : 4+0 : 2+0 : 2 c AIMA Slides � Stuart Russell and P eter Norvig, 1998 Chapter 14 13
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