artificial intelligence chapter 19 reasoning with ...19.5 uncertain evidence lwe must be certain...

Artificial Intelligence Artificial Intelligence Chapter 19Chapter 19

Reasoning with Uncertain Reasoning with Uncertain InformationInformation

Biointelligence LabSchool of Computer Sci. & Eng.

Seoul National University

OutlineOutline

l Review of Probability Theoryl Probabilistic Inferencel Bayes Networksl Patterns of Inference in Bayes Networks

(C) 1999-2009 SNU CSE Biointelligence Lab 2

l Patterns of Inference in Bayes Networksl Uncertain Evidencel D-Separationl Probabilistic Inference in Polytrees

19.1 Review of Probability Theory (1/4)19.1 Review of Probability Theory (1/4)

l Random variables

l Joint probabilitykVVV ,...,, 21

),...,,( 2211 kk vVvVvVp ===


(B (BAT_OK), M (MOVES) , L (LIFTABLE), G (GUAGE))

Joint Probability

(True, True, True, True) 0.5686(True, True, True, False) 0.0299(True, True, False, True) 0.0135(True, True, False, False) 0.0007… …

Ex.

),...,,( 2211 kk vVvVvVp ===

19.1 Review of Probability Theory (2/4)19.1 Review of Probability Theory (2/4)l Marginal probability

Ex. å=

==bB

GLMBpbBp ),,,()(

å==

===mMbB

GLMBpmMbBp,

),,,(),(


l Conditional probability

¨ Ex. The probability that the battery is charged given that the arm does not move

å==

===mMbB

GLMBpmMbBp,

),,,(),(

( ) ( )( )j

jiji Vp

VVpVVp

,| =

( ) ( )( )FalseMp

FalseMTrueBpFalseMTrueBp=

=====

,|



Figure 19.1 A Venn Diagram


l Chain rule

l Bayes’ rule

( ) ( ) ( ) ( ) ( )MpMGpMGLpMGLBpMGLBp |,|,,|,,, =

( ) ( )Õ=

-=k

iiik VVVpVVVp

11121 ,...,|,...,,


l Bayes’ rule

l set notation¨Abbreviation for

where

( ) ( ) ( )( )j

iijji Vp

VpVVpVVp

|| =

( )Vp( )kVVVp ,...,, 21

{ }kVVV ,...,, 21=V

19.2 Probabilistic Inference19.2 Probabilistic Inferencel We desire to calculate the probability of some variable Vi

has value vi given the evidence E =e.

[ ]

( ))(

3.0)(1.02.0

)(),,(),,(

)(),()|(

RpRp

RpRQPpRQPp

RpRQpRQp

Ø=

Ø+

=

ØØØ+Ø

=ØØ

=Ø

( ) ( )( )ep

eTrueVpeTrueVp ii =

=====

EE

E,|


p(P,Q,R) 0.3

p(P,Q,¬R) 0.2

p(P, ¬Q,R) 0.2

p(P, ¬Q,¬R) 0.1

p(¬P,Q,R) 0.05

p(¬P, Q, ¬R) 0.1

p(¬P, ¬Q,R) 0.05

p(¬P, ¬Q,¬R) 0.0

Example [ ]

( ))(

3.0)(1.02.0

)(),,(),,(

)(),()|(

RpRp

RpRQPpRQPp

RpRQpRQp

Ø=

Ø+

=

ØØØ+Ø

=ØØ

=Ø

[ ]

( ))(

1.0)(0.01.0

)(),,(),,(

)(),()|(

RpRp

RpRQPpRQPp

RpRQpRQp

Ø=

Ø+

=

ØØØØ+ØØ

=ØØØ

=ØØ

1)|()|(75.0)|(

=ØØ+Ø=Ø

RQpRQpRQp

Q

Statistical IndependenceStatistical Independence

l Conditional independence

¨ Intuition: Vi tells us nothing more about V than we already knew by knowing Vj

: a set of variables( ) ( ) ( )VVV |||, jiji VpVpVVp = V


j

l Mutually conditional independence

l Unconditional independence (When is empty)V

( ) ( )

( )Õ

Õ

=

=--

=

=

k

ii

k

iiiij

Vp

VVVVpVVVp

1

112121

|

,,...,,||,...,,

V

VV

( ) ( ) ( ) ( )kj VpVpVpVVVp ...,...,, 2121 =

19.3 Bayes Networks (1/2)19.3 Bayes Networks (1/2)

l Directed, acyclic graph (DAG) whose nodes are labeled by random variables

l Characteristics of Bayesian networks¨Node Vi is conditionally independent of any subset of


¨Node Vi is conditionally independent of any subset of nodes that are not descendents of Vi given its parents

l Prior probabilityl Conditional probability table (CPT)

( ) ( )Õ=

=k

iiik VPaVpVVVp

121 )(|,...,,

19.3 Bayes Networks (2/2)19.3 Bayes Networks (2/2)Bayes network about the block-lifting example


19.4 Patterns of Inference in Bayes Networks (1/3)19.4 Patterns of Inference in Bayes Networks (1/3)

l Causal or top-down inference¨ Ex. The probability that the arm moves given that the block is

liftableB

L

( ) ( ) ( )LBMpLBMpLMp |,|,| Ø+=


(chain rule)

(from the structure)

G M

0.9 * 0.95 0.855 .= =

( ) ( ) ( )LBMpLBMpLMp |,|,| Ø+=

( ) ( ) ( ) ( )LBpLBMpLBpLBMp |,||,| ØØ+=

( ) ( ) ( ) ( )BpLBMpBpLBMp ØØ+= ,|,|

l Diagnostic or bottom-up inference¨ Using an effect (or symptom) to infer a cause¨ Ex. The probability that the block is not liftable given that the arm

does not move.(using causal reasoning)

19.4 Patterns of Inference in Bayes Networks (2/3)19.4 Patterns of Inference in Bayes Networks (2/3)

B

G M

L

( ) 9525.0| =ØØ LMp


(using causal reasoning)

(using Bayes’ rule)

(using Bayes’ rule)

( ) 9525.0| =ØØ LMp

( ) ( ) ( )( ) ( ) ( )MpMpMp

LpLMpMLpØ

=Ø´

=Ø

ØØØ=ØØ

28575.03.09525.0||

( ) ( ) ( )( ) ( ) ( )MpMpMp

LpLMpMLpØ

=Ø´

=Ø

Ø=Ø

1015.07.0145.0||

( ) 7379.0| =ØØ MLp

l Explaining away¨ One evidence: ¬M (the arm does not move)¨ Additional evidence: ¬B (the battery is not charged)

(Bayes’ rule)

19.4 Patterns of Inference in Bayes Networks (3/3)19.4 Patterns of Inference in Bayes Networks (3/3)B

G M

L

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )( )

.30.0,

,|,

|,|,|,,|

=ØØ

ØØØØØ=

ØØØØØØØØ

=

ØØØØØØ

=ØØØ

MBpLpBpLBMp

MBpLpLBpLBMp

MBpLpLBMpMBLp


¨¬B explains ¬M, making ¬L less certain (0.30<0.7379)

(Bayes’ rule)

(def. of conditional prob.)

(structure of the Bayes network)

( ) ( ) ( )( )

( ) ( ) ( )( )

( ) ( ) ( )( )

.30.0,

,|,

|,|,|,,|

=ØØ

ØØØØØ=

ØØØØØØØØ

=

ØØØØØØ

=ØØØ

MBpLpBpLBMp

MBpLpLBpLBMp

MBpLpLBMpMBLp

19.5 Uncertain Evidence19.5 Uncertain Evidence

l We must be certain about the truth or falsity of the propositions they represent.¨ Each uncertain evidence node should have a child node, about

which we can be certain.¨ Ex. Suppose the robot is not certain that its arm did not move.


¨ Ex. Suppose the robot is not certain that its arm did not move.< Introducing M’ : “The arm sensor says that the arm moved”

– We can be certain that that proposition is either true or false.< p(¬L| ¬B, ¬M’) instead of p(¬L| ¬B, ¬M)

¨ Ex. Suppose we are uncertain about whether or not the battery is charged.< Introducing G : “Battery guage”< p(¬L| ¬G, ¬M’) instead of p(¬L| ¬B, ¬M’)

19.6 D19.6 D--Separation (1/3)Separation (1/3)

l D-saparation: direction-dependent separation

l Two nodes Vi and Vj are conditionally independent given a set of nodes E if for every undirected path in the Bayes network between Vi and Vj, there is some node, V , on the path having one of the following three


Ethere is some node, Vb, on the path having one of the following three properties.¨ Vb is in E, and both arcs on the path lead out of Vb¨ Vb is in E, and one arc on the path leads in to Vb and one arc leads out.¨ Neither Vb nor any descendant of Vb is in E, and both arcs on the path lead

in to Vb.l Vb blocks the path given E when any one of these conditions holds for

a path.l If all paths between Vi and Vj are blocked, we say that E d-separates Vi

and Vj


l Ex.¨ I(G, L|B) by rules 1 and 3

<By rule 1, B blocks the (only) path between G and L, given B.<By rule 3, M also blocks this path given B.

B

G M

L


¨ I(G, L)<By rule 3, M blocks the path between G and L.

¨ I(B, L)<By rule 3, M blocks the path between B and L.

l Even using d-separation, probabilistic inference in Bayes networks is, in general, NP-hard.

19.7 Probabilistic Inference in Polytrees (1/2)19.7 Probabilistic Inference in Polytrees (1/2)

l Polytree¨A DAG for which there is just one path, along arcs in

either direction, between any two nodes in the DAG.


l A node is above Q¨ The node is connected to Q only through Q’s parents

l A node is below Q¨ The node is connected to Q only through Q’s

19.7 Probabilistic Inference in Polytrees (2/2)19.7 Probabilistic Inference in Polytrees (2/2)


¨ The node is connected to Q only through Q’s immediate successors.

l Three types of evidences¨All evidence nodes are above Q.¨All evidence nodes are below Q.¨ There are evidence nodes both above and below Q.

Evidence Above (1/2)Evidence Above (1/2)

l Bottom-up recursive algorithml Ex. p(Q|P5, P4)( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )å

å

å

å

å

=

=

=

=

=

7,6

7,6

7,6

7,6

7,6

4|75|67,6|

4,5|74,5|67,6|

4,5|7,67,6|

4,5|7,64,5,7,6|

4,5|7,6,4,5|

PP

PP

PP

PP

PP

PPpPPpPPQp

PPPpPPPpPPQp

PPPPpPPQp

PPPPpPPPPQp

PPPPQpPPQp


( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )å

å

å

å

å

=

=

=

=

=

7,6

7,6

7,6

7,6

7,6

4|75|67,6|

4,5|74,5|67,6|

4,5|7,67,6|

4,5|7,64,5,7,6|

4,5|7,6,4,5|

PP

PP

PP

PP

PP

PPpPPpPPQp

PPPpPPPpPPQp

PPPPpPPQp

PPPPpPPPPQp

PPPPQpPPQp

(Structure of The Bayes network)

(d-separation)

(d-separation)

Evidence Above (2/2)Evidence Above (2/2)

l Calculating p(P7|P4) and p(P6|P5)( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

3 3

1, 2

7 | 4 7 | 3, 4 3 | 4 7 | 3, 4 3

6 | 5 6 | 1, 2 1 | 5 2P P

P P

p P P p P P P p P P p P P P p P

p P P p P P P p P P p P

= =

=

å åå


l Calculating p(P1|P5)¨ Evidence is “below”¨Here, we use Bayes’ rule

( ) ( ) ( )( )5

11|55|1Pp

PpPPpPPp =

Evidence Below (1/2)Evidence Below (1/2)

( ) ( ) ( )( )

( ) ( )( ) ( ) ( )

12, 13, 14, 11 || 12, 13, 14, 11

12, 13, 14, 11

12, 13, 14, 11 |

12, 13 | 14, 11 |

p P P P P Q p Qp Q P P P P

p P P P P

kp P P P P Q p Q

kp P P Q p P P Q p Q

=

=

=


l Using a top-down recursive algorithm( ) ( ) ( )12, 13 | 14, 11 |kp P P Q p P P Q p Q=

( ) ( ) ( )

( ) ( )å

å

=

=

9

9

|99|13,12

|9,9|13,12|13,12

P

P

QPpPPPp

QppQPPPpQPPp

( ) ( ) ( )å=8

8,8|9|9P

PpQPPpQPp ( ) ( ) ( )9|139|129|13,12 PPpPPpPPPp =

(d-separation)

Evidence Below (2/2)Evidence Below (2/2)

( ) ( ) ( )

( ) ( ) ( )å

å=

=

10

10

|1010|1110|14

|1010|11,14|11,14

P

P

QPpPPpPPp

QPpPPPpQPPp


( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

15

11

1

11 | 10 11 | 15, 10 15 | 10

15 | 10 15 | 10, 11 11

15, 10 | 11 1111 | 15, 10 15, 10 | 11 11

15, 10

15, 10 | 11 15 | 10, 11 10 | 11 15 | 10, 11 10

P

P

p P P p P P P p P P

p P P p P P P p P

p P P P p Pp P P P k p P P P p P

p P P

p P P P p P P P p P P p P P P p P

=

=

= =

= =

åå

Evidence Above and BelowEvidence Above and Below

( ) ( ) ( )| , |p Q p Q- + +

=E E E

( )}11,14,13,12{},4,5{| PPPPPPQpE+ E-


( ) ( ) ( )( )

( ) ( )( ) ( )

2

2

| , || ,

|

| , |

| |

p Q p Qp Q

p

k p Q p Q

k p Q p Q

- + +

+ -

- +

- + +

- +

=

=

=

E E EE E

E E

E E E

E E

(We have calculated two probabilities already)

(d-separation)

A Numerical Example (1/2)A Numerical Example (1/2)

( ) ( ) ( )QpQUkpUQp || =

•We want to calculate p(Q|U)

( )| ( | ) ( | )P

p U Q pU P p P Q=å

(Bayes’ rule)


( )| 0.6 0.05 0.03p Q U k k= ´ ´ = ´ To determine k, we need to calculate p(¬Q|U)

( ) ( ) ( )

( ) ( ) ( ) ( )80.099.08.001.095.0,|,|

,||

=´+´=ØØ+=

=åRpQRPpRpQRPp

RpQRPpQPpR

( ) 20.0| =Ø QPp

( ) ( ) ( )60.02.02.08.07.0

2.0|8.0||=´+´=

´Ø+´= PUpPUpQUp

A Numerical Example (2/2)A Numerical Example (2/2)

( ) ( ) ( )| |p Q U kp U Q p QØ = Ø Ø

( )| ( | ) ( | )P

p U Q pU P p P QØ = Øå

( ) ( ) ( )

( ) ( ) ( ) ( )019.099.001.001.090.0,|,|

,||

=´+´=ØØØ+Ø=

Ø=Ø åRpQRPpRpQRPp

RpQRPpQPpR

(Bayes’ rule)


Finally

( ) ( ) ( )

( ) ( ) ( ) ( )019.099.001.001.090.0,|,|

,||

=´+´=ØØØ+Ø=

Ø=Ø åRpQRPpRpQRPp

RpQRPpQPpR

( ) 98.0| =Ø QPp

( ) ( ) ( )21.098.02.019.07.0

98.0|019.0||=´+´=

´Ø+´=Ø PUpPUpQUp

( ) 20.095.021.0| ´=´´=Ø kkUQp ( )( )

( ) 13.003.035.4|,35.420.095.021.0|

03.005.06.0|

=´==\´=´´=Ø

´=´´=

UQpkkkUQp

kkUQp

Other methods for Probabilistic inference in Other methods for Probabilistic inference in Bayes NetworksBayes Networks

l Bucket elimination

l Monte Carlo methods (when the network is not a polytree)polytree)

l Clustering


Additional Readings (1/5)Additional Readings (1/5)

l [Feller 1968]¨ Probability Theory

l [Goldszmidt, Morris & Pearl 1990]¨Non-monotonic inference through probabilistic method


¨Non-monotonic inference through probabilistic method

l [Pearl 1982a, Kim & Pearl 1983]¨Message-passing algorithm

l [Russell & Norvig 1995, pp.447ff]¨ Polytree methods


l [Shachter & Kenley 1989]¨Bayesian network for continuous random variables

l [Wellman 1990]¨Qualitative networks


¨Qualitative networks

l [Neapolitan 1990]¨ Probabilistic methods in expert systems

l [Henrion 1990]¨ Probability inference in Bayesian networks


l [Jensen 1996]¨Bayesian networks: HUGIN system

l [Neal 1991]¨Relationships between Bayesian networks and neural


¨Relationships between Bayesian networks and neural networks

l [Hecherman 1991, Heckerman & Nathwani 1992]¨ PATHFINDER

l [Pradhan, et al. 1994]¨CPCSBN


l [Shortliffe 1976, Buchanan & Shortliffe 1984]¨MYCIN: uses certainty factor

l [Duda, Hart & Nilsson 1987]¨ PROSPECTOR: uses sufficiency index and necessity index


¨ PROSPECTOR: uses sufficiency index and necessity index

l [Zadeh 1975, Zadeh 1978, Elkan 1993]¨ Fuzzy logic and possibility theory

l [Dempster 1968, Shafer 1979]¨Dempster-Shafer’s combination rules


l [Nilsson 1986]¨ Probabilistic logic

l [Tversky & Kahneman 1982]¨Human generally loses consistency facing uncertainty


¨Human generally loses consistency facing uncertainty

l [Shafer & Pearl 1990]¨ Papers for uncertain inference

l Proceedings & Journals¨Uncertainty in Artificial Intelligence (UAI)¨ International Journal of Approximate Reasoning

artificial intelligence chapter 19 reasoning with ...19.5 uncertain evidence lwe must be certain...

Documents