Download Reasoning with Uncertain Information

Artificial Intelligence
Chapter 19
Reasoning with Uncertain
Information
Biointelligence Lab
School of Computer Sci. & Eng.
Seoul National University
Outline







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)

Random variables

Joint probability
Ex. (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
…
…
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19.1 Review of Probability Theory (2/4)

Marginal probability
Ex.

Conditional probability
 Ex. The probability that the battery is charged given
that the arm does not move
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19.1 Review of Probability Theory (3/4)
Figure 19.1 A Venn Diagram
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19.1 Review of Probability Theory (4/4)

Chain rule

Bayes’ rule

set notation
 Abbreviation for
where
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19.2 Probabilistic Inference

We desire to calculate the probability of some variable Vi
has value vi given the evidence E =e.
p (Q | ØR ) =
Example
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
=
p (Q , ØR )
p (ØR )
ép (P ,Q , ØR ) + p (ØP ,Q , ØR )ù
ê
ú
û
= ë
p (ØR )
(0.2 + 0.1) 0.3
=
p (ØR )
p (ØR )
p (ØQ | ØR ) =
p (ØQ , ØR )
p (ØR )
ép (P , ØQ , ØR ) + p (ØP , ØQ , ØR )ù
ê
ú
û
= ë
p (ØR )
(0.1 + 0.0) 0.1
=
=
p (ØR )
p (ØR )
p (Q | ØR ) = 0.75
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7R ) = 1
Statistical Independence

Conditional independence
: a set of variables
 Intuition: Vi tells us nothing more about V than we already knew
by knowing Vj

Mutually conditional independence

Unconditional independence
(When
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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 given its parents
k
pV1 , V2 ,...,Vk    pVi | Pa(Vi ) 
i 1

Prior probability
 Conditional probability table (CPT)
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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)

Causal or top-down inference
 Ex. The probability that the arm moves given that the block is
liftable
L
B
p (M | L ) = p (M , B | L ) + p (M , ØB | L )
G
M
= p (M | B , L )p (B | L ) + p (M | ØB , L )p (ØB | L ) (chain rule)
= p (M | B , L )p (B ) + p (M | ØB , L )p (ØB )
(from the structure)
= 0.9 * 0.95 = 0.855 .
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19.4 Patterns of Inference in Bayes Networks (2/3)
L

Diagnostic or bottom-up inference
B
 Using an effect (or symptom) to infer a cause
G
M
 Ex. The probability that the block is not liftable given that the arm
does not move.
pM | L  0.9525 (using causal reasoning)
p (ØL | ØM ) =
p (L | ØM ) =
p (ØM | ØL )p (ØL )
p (ØM )
p (ØM | L )p (L )
p (ØM )
=
=
0.9525 ´ 0.3 0.28575
=
p (ØM )
p (ØM )
0.145 ´ 0.7
0.1015
=
p (ØM )
p (ØM )
(using Bayes’ rule)
(using Bayes’ rule)
p (ØL | ØM ) = 0.7379
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19.4 Patterns of Inference in Bayes Networks (3/3)
L
B

Explaining away
G
 One evidence: ØM (the arm does not move)
 Additional evidence: ØB (the battery is not charged)
p (ØL | ØB , ØM ) =
=
=
p (ØM , ØB | ØL )p (ØL )
p (ØB , ØM )
p (ØM | ØB , ØL )p (ØB | ØL )p (ØL )
p (ØB , ØM )
p (ØM | ØB , ØL )p (ØB )p (ØL )
p (ØB , ØM )
M
(Bayes’ rule)
(def. of conditional prob.)
(structure of the
Bayes network)
= 0.30.
 ¬B explains ¬M, making ¬L less certain (0.30<0.7379)
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19.5 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
which 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)

D-saparation: direction-dependent separation

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, 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.


Vb blocks the path given E when any one of these conditions holds for
a path.
If all paths between Vi and Vj are blocked, we say that E d-separates Vi
and Vj
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19.6 D-Separation (2/3)
Figure 19.3 Conditional Independence via Blocking Nodes
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19.6 D-Separation (3/3)
L
B

Ex.
 I(G, L|B) by rules 1 and 3
G
M
 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 Polytrees (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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19.7 Probabilistic Inference in Polytrees (2/2)

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 evidences
 All evidence nodes are above Q.
 All evidence nodes are below Q.
 There are evidence nodes both above and below Q.
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Evidence Above (1/2)

Bottom-up recursive algorithm
 Ex. p(Q|P5, P4)
pQ | P5, P 4 
 pQ, P6, P7 | P5, P4
P 6, P 7

 pQ | P6, P7, P5, P4 pP6, P7 | P5, P4
P 6, P 7

 pQ | P6, P7 pP6, P7 | P5, P4
P 6, P 7

 pQ | P6, P7 pP6 | P5, P4 pP7 | P5, P4
P 6, P 7

 pQ | P6, P7 pP6 | P5 pP7 | P4
(Structure of
The Bayes network)
(d-separation)
(d-separation)
P 6, P 7
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Evidence Above (2/2)

Calculating p(P7|P4) and p(P6|P5)
p (P 7 | P 4) = å p (P 7 | P 3, P 4)p (P 3 | P 4) = å p (P 7 | P 3, P 4)p (P 3)
p (P 6 | P 5) = å p (P 6 | P 1, P 2)p (P 1 | P 5)p (P 2)
P3
P3
P 1,P 2

Calculating p(P1|P5)
 Evidence is “below”
 Here, we use Bayes’ rule
pP5 | P1 pP1
pP1 | P5 
pP5
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Evidence Below (1/2)
p (Q | P 12, P 13, P 14, P 11) =
p (P 12, P 13, P 14, P 11 | Q )p (Q )
p (P 12, P 13, P 14, P 11)
= kp (P 12, P 13, P 14, P 11 | Q )p (Q )
= kp (P 12, P 13 | Q )p (P 14, P 11 | Q )p (Q )

Using a top-down recursive algorithm
pP12, P13 | Q    pP12, P13 | P9, Q  p p9 | Q 
P9
  pP12, P13 | P9  pP9 | Q 
(d-separation)
P9
pP9 | Q    pP9 | P8, Q  pP8
pP12, P13 | P9  pP12 | P9 pP13 | P9
P8
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Evidence Below (2/2)
pP14, P11 | Q    pP14, P11 | P10 pP10 | Q 
P10
  pP14 | P10 pP11 | P10 pP10 | Q 
P10
å p (P 11 | P 15, P 10)p (P 15 | P 10)
p (P 15 | P 10) = å p (P 15 | P 10, P 11)p (P 11)
p (P 15, P 10 | P 11)p (P 11)
p (P 11 | P 15, P 10) =
= k p (P 15, P 10 | P 11)p (P 11)
p (P 15, P 10)
p (P 15, P 10 | P 11) = p (P 15 | P 10, P 11)p (P 10 | P 11) = p (P 15 | P 10, P 11)p (P 10)
p (P 11 | P 10) =
P 15
P 11
1
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Evidence Above and Below
pQ | {P5, P4},{P12, P13, P14, P11}
E-
E+
(
+
-
p Q | E ,E
(
)
(
)
)=
p (E | E )
= k p (E | Q , E ) p (Q | E )
= k p (E | Q ) p (Q | E )
(d-separation)
p E- | Q , E+ p Q | E+
-
-
+
+
+
2
-
+
2
(We have calculated two probabilities already)
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A Numerical Example (1/2)
• We want to calculate p(Q|U)
pQ | U   kpU | Q pQ (Bayes’ rule)
p (U | Q ) =
å
p(U | P )p(P | Q )
P
p  P | Q    p  P | R, Q  p  R 
R
 pP | R, Q  pR   pP | R, Q  pR 
p (ØP | Q ) = 0.20
 0.95  0.01  0.8  0.99  0.80
pU | Q  pU | P  0.8  pU | P  0.2
 0.7  0.8  0.2  0.2  0.60
p (Q | U ) = k ´ 0.6 ´ 0.05 = k ´ 0.03
To determine k, we need to calculate p(¬Q|U)
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A Numerical Example (2/2)
p (ØQ | U ) = kp (U | ØQ )p (ØQ )
(Bayes’ rule)
p (U | ØQ ) =
å
p(U | P )p(P | ØQ )
P
p (P | ØQ ) =
å p (P | R , ØQ )p (R )
p (P | R , ØQ )p (R ) + p (P | ØR , ØQ )p (ØR )
R
=
= 0.90 ´ 0.01 + 0.01 ´ 0.99 = 0.019
p (ØP | ØQ ) = 0.98
p (U | ØQ ) = p (U | P )´ 0.019 + p (U | ØP )´ 0.98
= 0.7 ´ 0.19 + 0.2 ´ 0.98 = 0.21
Finally
pQ | U   k  0.6  0.05  k  0.03
p (ØQ | U ) = k ´ 0.21 ´ 0.95 = k ´ 0.20
pQ | U   k  0.21 0.95  k  0.20
 k  4.35, pQ | U   4.35  0.03  0.13
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Other methods for Probabilistic inference in
Bayes Networks

Bucket elimination

Monte Carlo methods (when the network is not a
polytree)

Clustering
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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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Additional Readings (2/5)

[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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Additional Readings (3/5)

[Jensen 1996]
 Bayesian networks: HUGIN system

[Neal 1991]
 Relationships between Bayesian networks and neural
networks

[Hecherman 1991, Heckerman & Nathwani 1992]
 PATHFINDER

[Pradhan, et al. 1994]
 CPCSBN
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Additional Readings (4/5)

[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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Additional Readings (5/5)

[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
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