Bernal S D_2010.pdf - University of Plymouth

3 J. DEFINITION AND MATHEMATICAL FORMULATION
of questions they need to answer Instead of asking a question for each of the combinations of
parental states, the model assumes that for each state of each parent node, the rest of the parents
are in a compatible or most-likely state.
Previous models, such as the Noisy-OR, are constrained by the assumption that parents act
independently without synergy. This means parents individually influence the child and that
there are negligible cross-interactions between individual pareni-lo-child inlliiences, such as in
the Wave*-example.
However, this method is derived for a different class of Bayesian networks in which there nec­
essarily exists a coherent frame of knowledge where the effect is a result of the interactions be­
tween the parents. An excellent example is provided by Das (2004), where the efficiency (E) of a
company is affected by three parent nodes; personnel nwralc (VM.).personnel training (PT)and
managerial expertise (ME), with states very-low.low,average,high,very-high. Clearly these
causes are related, such that a possible compatible parental configuration when the personnel
morale is high is {Comp{PM = high)} — {PM = high.PT — high,ME = high]. This means
when the personnel morale is high, it is also likely that the personnel training and the manage­
rial expertise are high.
More formally, given a node X with a set of parents IJ\, ...,lJs, the stale Uj = uj is compatible
with the state IJj ~ «,-, if according to the expert's mental model the state Uj — Uj is most
likely to coexist with the state Ui =• «,-. Let {CompiUi = M,)} denote the compatible parental
configuration where U, is in the state », and the rest of the parents are in states compatible with
Ui = «/•
For each compatible parental conhguration it is now possible to calculate the conditional prob­
ability distribution over the .slates of the child node X in ihc form
PiX = 0\{Comp{U, = Ui)}),P{X = ]\{Comp{U^ = «,•)}) P{X - kx\{Comp(U< - «;)})
for I = 1 ,.A' and u, = i-.ka,, where kx is ihe number of slates of X, and ky, is the number of
states of parent node U,.
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