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

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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 ===

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(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,

),,,(),(

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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=

=====

,|

19.1 Review of Probability Theory (3/4)19.1 Review of Probability Theory (3/4)

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Figure 19.1 A Venn Diagram

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

l Chain rule

l Bayes’ rule

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

( ) ( )Õ=

-=k

iiik VVVpVVVp

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

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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,|

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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

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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

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¨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

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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+=

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(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

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(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

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¨¬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.

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¨ 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

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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

19.6 D19.6 D--Separation (2/3)Separation (2/3)

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

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

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¨ 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.

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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)

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¨ 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

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( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( )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

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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

=

=

=

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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

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

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

( ) ( ) ( ) ( ) ( )

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-

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( ) ( ) ( )( )

( ) ( )( ) ( )

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)

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( )| 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)

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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

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

l [Feller 1968]¨ Probability Theory

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

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¨Non-monotonic inference through probabilistic method

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

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

Additional Readings (2/5)Additional Readings (2/5)

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

l [Wellman 1990]¨Qualitative networks

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¨Qualitative networks

l [Neapolitan 1990]¨ Probabilistic methods in expert systems

l [Henrion 1990]¨ Probability inference in Bayesian networks

Additional Readings (3/5)Additional Readings (3/5)

l [Jensen 1996]¨Bayesian networks: HUGIN system

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

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¨Relationships between Bayesian networks and neural networks

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

l [Pradhan, et al. 1994]¨CPCSBN

Additional Readings (4/5)Additional Readings (4/5)

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

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

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¨ 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

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

l [Nilsson 1986]¨ Probabilistic logic

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

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¨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