Artificial Intelligence Chapter 19 Reasoning with Uncertain Information Biointelligence Lab School of Computer Sci. & Eng. Seoul National University

Artificial Intelligence Artificial Intelligence Chapter 19Chapter 19

Reasoning with Uncertain Reasoning with Uncertain InformationInformation

Biointelligence Lab

School of Computer Sci. & Eng.

Seoul National University

(C) 2000-2002 SNU CSE Biointelligence Lab

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OutlineOutline

Review of Probability Theory Probabilistic Inference Bayes Networks Patterns of Inference in Bayes Networks Uncertain Evidence D-Separation Probabilistic Inference in Polytrees


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

Random variables

Joint probability

(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.


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

Conditional probability

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

Ex.


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Figure 19.1 A Venn Diagram


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Chain rule

Bayes’ rule

Abbreviation for where


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19.2 Probabilistic Inference19.2 Probabilistic Inference

The probability some variable Vi has value vi given the evidence =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

RpRp

Rp

RQPpRQPp

Rp

RQpRQp

3.01.02.0

,,),,,|

RpRp

Rp

RQPpRQPp

Rp

RQpRQp

1.00.01.0

,,),,,|

1|||

75.0|

RQpRQP

RQp


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Statistical IndependenceStatistical Independence

Conditional independence

Mutually conditional independence

Unconditional independence


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

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

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

nodes that are not descendents of Vi.

Prior probability Conditional probability table (CPT)

k

iiik VPaVpVVVp

121 )(|,...,,


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


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19.4 Patterns of Inference in Bayes 19.4 Patterns of Inference in Bayes Networks (1/3)Networks (1/3) Causal or top-down inference

Ex. The probability that the arm moves given that the block is liftable

BpLBMpBpLBMp

LBpLBMpLBpLBMp

LBMpLBMpLMp

,|,|

|,||,|

|,|,|


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

9525.0| LMp (using a causal reasoning)

88632.0|

03665.07.00595.0||

28575.03.09525.0||

MLp

MpMpMp

LpLMpMLp

MpMpMp

LpLMpMLp (Bayes’ rule)


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19.4 Patterns of Inference in Bayes 19.4 Patterns of Inference in Bayes Networks (3/3)Networks (3/3) Explaining away

¬B explains ¬M, making ¬L less certain

88632.0030.0

,

,|

,

|,|

,

|,,|

MBp

LpBpLBMp

MBp

LpLBpLBMp

MBp

LpLBMpMBLp (Bayes’ rule)

(def. of conditional prob.)

(structure of the Bayes network)


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19.5 Uncertain Evidence19.5 Uncertain Evidence

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

ch we can be certain. 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’)


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19.6 D-Separation (1/3)19.6 D-Separation (1/3)

d-separates Vi and Vj if for every undirected path in the Bayes network between Vi and Vj, there is some node, Vb, on the path having one of the following three properties. Vb is in , and both arcs on the path lead out of Vb

Vb is in , and one arc on the path leads in to Vb and one arc leads out.

Neither Vb nor any descendant of Vb is in , and both arcs on the path lead in to Vb.

Vb blocks the path given when any one of these conditions holds for a path.

Two nodes Vi and Vj are conditionally independent given a set of node


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Figure 19.3 Conditional Independence via Blocking Nodes


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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.

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.

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


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19.7 Probabilistic Inference in Polytre19.7 Probabilistic Inference in Polytrees (1/2)es (1/2) 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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A node is above Q The node is connected to Q only through Q’s parents

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

immediate successors.

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

19.7 Probabilistic Inference in Polytre19.7 Probabilistic Inference in Polytrees (2/2)es (2/2)


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

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

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)


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

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

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

2,1

3 3

25|12,1|65|6

34,3|74|34,3|74|7

PP

P P

PpPPpPPPpPPp

PpPPPpPPpPPPpPPp

5

11|55|1

Pp

PpPPpPPp


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

Top-down recursive algorithm

QpQPPpQPPkp

QpQPPPPkp

PPPPp

QpQPPPPpPPPPQp

|11,14|13,12

|11,14,13,12

11,14,13,12

|11,14,13,121,14,13,12|

9

9

|99|13,12

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

P

P

QPpPPPp

QppQPPPpQPPp

8

8,8|9|9P

PpQPPpQPp 9|139|129|13,12 PPpPPpPPPp


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

10

10

|1010|1110|14

|1010|11,14|11,14

P

P

QPpPPpPPp

QPpPPPpQPPp

1011,10|1511|1011,10|1511|10,15

1111|10,1510,15

1111|10,1510,15|11

1111,10|1510|15

10|1510,15|1110|11

1

11

15

PpPPPpPPpPPPpPPPp

PpPPPpkPPp

PpPPPpPPPp

PpPPPpPPp

PPpPPPpPPp

P

P


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Evidence Above and BelowEvidence Above and Below

||

|,|

|

|,|,|

2

2

QpQpk

QpQpk

p

QpQpQp

}11,14,13,12{},4,5{| PPPPPPQp+ -


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

80.099.08.001.095.0

,|,|

,||

RpQRPpRpQRPp

RpQRPpQPpR

019.099.001.001.090.0

,|,|

,||

RpQRPpRpQRPp

RpQRPpQPpR


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

Other techniques Bucket elimination Monte Carlo method Clustering

60.02.02.08.07.0

2.0|8.0||

PUpPUpQUp

21.098.02.0019.07.0

98.0|019.0||

PUpPUpQUp

13.003.035.4|,35.4

20.095.021.0|

03.005.06.0|

UQpk

kkUQp

kkUQp


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

[Feller 1968] Probability Theory

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

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

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


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

[Wellman 1990] Qualitative networks

[Neapolitan 1990] Probabilistic methods in expert systems

[Henrion 1990] Probability inference in Bayesian networks


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[Jensen 1996] Bayesian networks: HUGIN system

[Neal 1991] Relationships between Bayesian networks and neural n

etworks

[Hecherman 1991, Heckerman & Nathwani 1992] PATHFINDER

[Pradhan, et al. 1994] CPCSBN


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

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

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

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


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[Nilsson 1986] Probabilistic logic

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

[Shafer & Pearl 1990] Papers for uncertain inference

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

Documents

Artificial Intelligence Chapter 19 Reasoning with Uncertain Information Biointelligence Lab School of Computer Sci. & Eng. Seoul National University