[PPT] - Uncertain t y Chapter 14 c AIMA Slides Stuart Russell and PowerPoint Presentation

SLIDE 1 Uncertain t y Chapter 14 AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 1

SLIDE 2 Outline } Uncertaint y } Probabili t y } Syntax } Semantics } Inference rules AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 2

SLIDE 3 Uncertain t y Let action A t = leave fo r airp

rt

t minutes b efo re ight Will A t get me there

n

time? Problems: 1) pa rtial

bservabili

t y (road state,

ther

drivers' plans, etc.) 2) noisy senso rs (K CBS trac rep

rts)

3) uncertain t y in action

utcomes

(at tire, etc.) 4) immense complexit y

f

mo delling and p redicting trac Hence a purely logical app roach either 1) risks falseho

d:

\A 25 will get me there

n

time"

r

2) leads to conclusions that a re to

w

eak fo r decision making: \A 25 will get me there

n

time if there's no accident

n

the b ridge and it do esn't rain and my tires remain intact etc etc." (A 1440 might reasonably b e said to get me there

n

time but I'd have to sta y

vernight

in the airp

rt

: : :) AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 3

SLIDE 4 Metho ds for handling uncertain t y Default

r

nonmonotonic logic: Assume my ca r do es not have a at tire Assume A 25 w

rks

unless contradicted b y evidence Issues: What assumptions a re reasonable? Ho w to handle contradicti

n?

Rules with fudge facto rs : A 25 7! 0:3 get there

n

time S pr ink l er 7! 0:99 W etGr ass W etGr ass 7! 0:7 R ain Issues: Problems with combinati

n,

e.g., S pr ink l er causes R ain?? Probabili t y Given the available evidence, A 25 will get me there

n

time with p robabili t y 0.04 Mahaviraca ry a (9th C.), Ca rdamo (1565) theo ry

f

gambling (F uzzy logic handles de gr e e

f

truth NOT uncertaint y e.g., W etGr ass is true to degree 0.2) AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 4

SLIDE 5 Probabilit y Probabili stic assertions summarize eects

f

laziness : failure to enumerate exceptions, qualicati

ns,

etc. igno rance : lack

f

relevant facts, initial conditions, etc. Subjective

r

Ba y esian p robabili t y: Probabili ties relate p rop

sitions

to

ne's
wn

state

f

kno wledge e.g., P (A 25 jno rep

rted

accidents ) = 0:06 These a re not assertions ab

ut

the w

rld

Probabili ties

f

p rop

sitions

change with new evidence: e.g., P (A 25 jno rep

rted

accidents ; 5 a.m.) = 0:15 (Analogous to logical entailment status K B j = , not truth.) AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 5

SLIDE 6 Making decisions under uncertain t y Supp

se

I b elieve the follo wing: P (A 25 gets me there

n

time j : : :) = 0:04 P (A 90 gets me there

n

time j : : :) = 0:70 P (A 120 gets me there

n

time j : : :) = 0:95 P (A 1440 gets me there

n

time j : : :) = 0:9999 Which action to cho

se?

Dep ends

n

my p references fo r missing ight vs. airp

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 AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 6

SLIDE 7 Axioms

f

probabilit y F

r

any p rop

sitions

A, B 1.

P

(A)

1

2. P (T r ue) = 1 and P (F al se) = 3. P (A _ B ) = P (A) + P (B )

P

(A ^ B )

>

A B True 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

f
utcome.

AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 7

SLIDE 8 Syn tax Simila r to p rop

sitional

logic: p

ssible

w

rlds

dened b y assignment

f

values to random va riables . Prop

sitiona

l

r

Bo

lean

random va riables e.g., C av ity (do I have a cavit y?) Include p rop

sitiona

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

ne
f

hsunny ; r ain; cl

udy

; snow i V alues must b e exhaustive and mutually exclusive Prop

sition

constructed b y assignment

f

a value: e.g., W eather = sunny ; also C av ity = tr ue fo r cla rit y AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 8

SLIDE 9 Syn tax con td. Prio r

r

uncondition al p robabili tie s

f

p rop

sition

s e.g., P (C av ity ) = 0:1 and P (W eather = sunny ) = 0:72 co rresp

nd

to b elief p rio r to a rrival

f

any (new) evidence Probabili t y distribution gives values fo r all p

ssible

assignments: P(W eather ) = h0:72; 0:1; 0:08; 0:1i (no rmalize d , i.e., sums to 1) Joint p robabilit y distributi

n

fo r a set

f

va riables gives values fo r each p

ssible

assignment to all the va riables P(W eather ; C av ity ) = a 4

2

matrix

f

values: W eather = sunny r ain cl

udy

snow C av ity = tr ue C av ity = f al se AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 9

SLIDE 10 Syn tax con td. Conditional

r

p

sterio

r p robabili tie s e.g., P (C av ity jT

othache)

= 0:8 i.e., given that T

othache

is all I kno w Notation fo r conditiona l distributi

ns:

P(W eather jE ar thq uak e) = 2-element vecto r

f

4-element vecto rs If w e kno w mo re, e.g., C av ity is also given, then w e have P (C av ity jT

othache;

C av ity ) = 1 Note: the less sp ecic 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 simplicati

n,

e.g., P (C av ity jT

othache;

49er sW in) = P (C av ity jT

othache)

= 0:8 This kind

f

inference, sanctioned b y domain kno wledge, is crucial AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 10

SLIDE 11 Conditional probabilit y Denition

f

conditional p robabili t y: P (AjB ) = P (A ^ B ) P (B ) if P (B ) 6= Pro duct rule gives an alternative fo rmulation: P (A ^ B ) = P (AjB )P (B ) = P (B jA)P (A) A general version holds fo r whole distribution s, e.g., P(W eather ; C av ity ) = P(W eather jC av ity )P (C av ity ) (View as a 4

2

set

f

equations, not matrix mult.) Chain rule is derived b y successive applicati

n
f

p ro duct rule: P(X 1 ; : : : ; X n ) = P(X 1 ; : : : ; X n1 ) P(X n jX 1 ; : : : ; X n1 ) = P (X 1 ; : : : ; X n2 ) P(X n 1 jX 1 ; : : : ; X n2 ) P (X n jX 1 ; : : : ; X n1 ) = : : : =

n

i = 1 P(X i jX 1 ; : : : ; X i1 ) AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 11

SLIDE 12 Ba y es' Rule Pro duct rule P (A ^ B ) = P (AjB )P (B ) = P (B jA)P (A) ) Ba y es' rule P (AjB ) = P (B jA)P (A) P (B ) Why is this useful??? F

r

assessing diagnostic p robabili t y from causal p robabilit y: P (C ausejE f f ect) = P (E f f ectjC ause)P ( C aus e) P (E f f ect) E.g., let M b e meningitis, S b e sti neck: P (M jS ) = P (S jM )P (M ) P (S ) = 0:8

0:0001

0:1 = 0:0008 Note: p

sterio

r p robabilit y

f

meningiti s still very small! AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 12

SLIDE 13 Normalization Supp

se

w e wish to compute a p

sterio

r distributi

n
ver

A given B = b, and supp

se

A has p

ssible

values a 1 : : : a m W e can apply Ba y es' rule fo r each value

f

A: P (A = a 1 jB = b) = P (B = bjA = a 1 )P (A = a 1 )=P (B = b) : : : P (A = a m jB = b) = P (B = bjA = a m )P (A = a m )=P (B = b) Adding these up, and noting that

i

P (A = a i jB = b) = 1: 1=P (B = b) = 1= i P (B = bjA = a i )P (A = a i ) This is the no rmalizati

n

facto r , constant w.r.t. i, denoted : P (AjB = b) = P(B = bjA)P (A) T ypically compute an unno rmalize d distributi

n,

no rmalize at end e.g., supp

se

P (B = bjA)P(A) = h0:4; 0:2; 0:2i then P(AjB = b) = h0:4; 0:2; 0:2i = h0:4;0:2;0:2i 0:4+0:2+0:2 = h0:5; 0:25; 0:25i AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 13

SLIDE 14 Conditioning Intro ducin g a va riable as an extra condition: P (X jY ) =

z

P (X jY ; Z = z )P (Z = z jY ) Intuition:

ften

easier to assess each sp ecic circumstance , e.g., P (R unO v er jC r

ss)

= P (R unO v er jC r

ss;

Lig ht = g r een)P (Lig ht = g r eenjC r

ss)

+ P (R unO v er jC r

ss;

Lig ht = y el l

w

)P (Lig ht = y el l

w

jC r

ss)

+ P (R unO v er jC r

ss;

Lig ht = r ed)P (Lig ht = r edjC r

ss)

When Y is absent, w e have summing

ut
r

ma rginaliz ati

n

: P (X ) =

z

P (X jZ = z )P (Z = z ) =

z

P (X ; Z = z ) In general, given a joint distributi

n
ver

a set

f

va riables, the distribution

ver

any subset (called a ma rginal distribution fo r histo rical reasons) can b e calculate d b y summing

ut

the

ther

va riables. AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 14

SLIDE 15 F ull join t distributions A complete p robabili t y mo del sp ecies every entry in the joint distribution fo r all the va riables X = X 1 ; : : : ; X n I.e., a p robabilit y fo r each p

ssible

w

rld

X 1 = x 1 ; : : : ; X n = x n (Cf. complete theo ries in logic.) E.g., supp

se

T

othache

and C av ity a re the random va riables: T

othache

= tr ue T

othache

= f al se C av ity = tr ue 0:04 0:06 C av ity = f al se 0:01 0:89 P

ssible

w

rlds

a re mutually exclusive ) P (w 1 ^ w 2 ) = P

ssible

w

rlds

a re exhaustive ) w 1 _

_

w n is T r ue hence

i

P (w i ) = 1 AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 15

SLIDE 16 F ull join t distributions con td. 1) F

r

any p rop

sition
dened
n

the random va riables (w i ) is true

r

false 2)

is

equivalen t to the disjunction

f

w i s where (w i ) is true Hence P () =

fw

i : (w i )g P (w i ) I.e., the unconditi

na

l p robabilit y

f

any p rop

sition

is computable as the sum

f

entries from the full joint distributi

n

Conditional p robabili ti es can b e computed in the same w a y as a ratio: P (j ) = P ( ^

)

P ( ) E.g., P (C av ity jT

othache)

= P (C av ity ^ T

othache)

P (T

othache)

= 0:04 0:04 + 0:01 = 0:8 AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 16

SLIDE 17 Inference from join t distributions T ypically , w e a re interested in the p

sterio

r joint distribut ion

f

the query va riables Y given sp ecic values e fo r the evidence va riables E Let the hidden va riables b e H = X

Y
E

Then the required summation

f

joint entries is done b y summing

ut

the hidden va riables: P (Y jE = e ) = P(Y; E = e ) =

h

P(Y; E = e ; H = h) The terms in the summation a re joint entries b ecause Y, E, and H together exhaust the set

f

random va riables Obvious p roblems: 1) W

rst-case

time complexit y O (d n ) where d is the la rgest a rit y 2) Space complexit y O (d n ) to sto re the joint distribut ion 3) Ho w to nd the numb ers fo r O (d n ) entries??? AIMA Slides c Stuart Russell and P eter Norvig, 1998 Chapter 14 17