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36 Janneke H. Bolt and Linda C. van der Gaag
z
1
0.78
0.54
0
x
1 0
Figure 3: The line segment capturing Pr(c) and the
surface capturing f Pr(c) as a function of x = πC(a)
and y = πC(b) for the network from Figure 1.
y
1
z
1
0.441
0.207
0
x
1 0
Figure 4: The line segment capturing Pr(c | f) and
the surface capturing f Prconv(c | f) as a function of
x = πC(a) and y = πC(b) for the network from
Figure 1.
account by Pearl’s propagation algorithm. As a result, the difference between the exact probability Pr(c),
and the approximate probability � Pr(c) calculated by Pearl’s algorithm equals
Pr(c) − � Pr(c) = � Pr(c | ab) − Pr(c | a ¯ b) − Pr(c | āb) + Pr(c | ā ¯ b) � ·
� Pr(a | d) − Pr(a | ¯ d) � ·
� Pr(b | d) − Pr(b | ¯ d) � ·
� Pr(d) − Pr(d) 2 �
In the sequel, v is used to denote the prior convergence error Pr(c) − � Pr(c). The absolute value of the v
ranges between 0 and 0.5.
The prior convergence error, is illustrated in Figure 3; for the construction of the figure we used the
network from Figure 1. The line segment captures the exact probability z = Pr(c) as a function of Pr(d),
given the conditional probabilities as specified for the nodes A, B and C of the example network. Note that
each specific Pr(d) corresponds with a unique combination of x = πC(a) and y = πC(b). The depicted
surface captures z = � Pr(c) as a function of x = πC(a) and y = πC(b), given the conditional probabilities
as specified for C of the example network. The approximate probability � Pr(c) is computed from the exact
messages πC(a) and πC(b), and therefore, the convergence error equals the distance between the point on
the line segment that matches the probability Pr(d) from the network and its orthogonal projection on the
surface. For the Pr(d) = 0.5 from the network, the difference between Pr(c) and � Pr(c) is indicated by the
vertical dotted line segment and equals 0.78 − 0.54 = 0.24.
4 The Posterior Error at Convergence Nodes
In this section the posterior error found in the probabilities computed for the convergence node in a network
with just a simple loop is investigated. In Section 4.1 we thereby abstract from the cycling error by considering
the error which would be found given that the convergence node receives the exact causal messages
from its parent nodes. Thereafter, in Section 4.2, the effect of the cycling error in the causal messages on
the error found in the probabilities computed for the convergence node is discussed.
4.1 The Posterior Convergence Error
The error that has arised in the probabilities computed for a convergence node of a Bayesian network in
its prior state, may change in size as soon as an observation is entered into the network. An observation
can influence the computed probabilities through causal messages or through diagnostic messages to the
convergence node. An observation that affects the error through a causal message trivially changes the prior
convergence error by conditioning all probabilities involved on the entered observation. An observation that
affects the error through a diagnostic message, on the other hand, fundamentally changes the expression of
y
1