Slide 8.1
8
REASONING IN UNCERTAIN
SITUATIONS
8.0
Introduction
8.1
Logic-Based Abductive Inference
8.2
Abduction: Alternatives to Logic
8.3
The Stochastic Approach to
Uncertainty
8.4
Epilogue and References
8.5
Exercises
Slide 8.2
Figure 8.1: A justification network to believe that Peter studies hard.
Figure 8.2: 8.2(a) is a premise justification, and 8.2(b) the ANDing of two
beliefs, a and not b, to support c (Goodwin 1982).
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.3
Figure 8.3: The new labeling of Figure 8.1 associated with the new premise
party_person(david).
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.4
Figure 8.4: An ATMS labeling of nodes in a dependency network.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.5
Figure 8.5: The lattice for the premises of the network of Figure 8.4. Circled
sets indicate the hierarchy of inconsistencies, after Martins (1991).
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.6
Figure 8.6: The fuzzy set representation for “small integers.”
Figure 8.7
A fuzzy set representation for the sets short, median, and tall
males.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.7
Figure 8.8: The inverted pendulum and the angle θ and dθ/dt input values.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Figure 8.9: The fuzzy regions for the input values θ (a) and dθ/dt (b).
Slide 8.8
Figure 8.10 The fuzzy regions of the output value u, indicating the movement
of the pendulum base.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.9
Figure 8.11: The fuzzification of the input measures x1=1, x2 = -4.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.10
Figure 8.12: The Fuzzy Associative Matrix (FAM) for the pendulum problem.
The input values are on the left and top.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
(b). The centroid of the union (-2) is the
crisp output.
Figure 8.13: The fuzzy consequents (a) and their union
Slide 8.11
© 2002 Addison Wesley
Slide 8.12
Minimum of their measures is taken as the measure of the rule result:
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.13
Table 8.1:
Using Dempster’s rule to obtain a belief distribution for m3.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.14
Table 8.2:
Using Dempster’s rule to combine m3 and m4 to get m5.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.15
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.16
Probability theory, the general form of Bayes’ theorem
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.17
Figure 8.14: The Bayesian representation of the traffic problem with potential
explanations.
Table 8.3
The joint probability distribution for the traffic and construction
variables of Figure 8.14.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.18
Figure 8.15: Figure 8.15a is a serial connection of nodes where influence
runs between A and B unless V is instantiated. Figure 8.15b is a
diverging connection, where influence runs between V’s children,
unless V is instantiated. In Figure 8.15c, a converging connection,
if nothing is known about V then its parents are independent,
otherwise correlations exist between its parents.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Defining the d-separation of nodes in a belief network (after Pearl 1988)
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
Slide 8.19
© 2002 Addison Wesley
Slide 8.20
Figure 8.16: An example of a Bayesian probabilistic network, where the
probability dependencies are located next to each node.
This example is from Pearl (1988).
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Table 8.4:
The probability distribution for P(WS), a function of P(W) and
P(R), given the effect of S. We calculate the effect for x, where
R = t and W = t.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
Slide 8.21
© 2002 Addison Wesley
Slide 8.22
Figure 8.17: A junction tree (a) for the Bayesian probabilistic network of
(b). Note that we started to construct the transition table for
the rectangle R,W.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley
Slide 8.23
Figure 8.18: A belief network representing the possibility of a house alarm
in a dangerous neighborhood.
A R T I F I C I A L I N T E L L I G E N C E: Structure and Strategies for Complex Problem Solving, 4th Edition George F. Luger
© 2002 Addison Wesley