Stefan Growe
hypotheses only. A judgement calculus that considers exclusively the evidence, like the possibilistic one described earlier,
produces identical judgements for all solutions. Based on the conditional probabilities of the state transitions it would be
most promising, for the given example, to prefer the successor state FairInactivity because it is the most probable one.
In order to integrate prior probabilities of object states and their transitions within the judgement of a scene description,
a probabilistic approach is suggested, which is described in the following.
4 PROBABILISTIC JUDGEMENT CALCULUS
The developed probabilistic judgement approach transforms the semantic net, i.e. the scene interpretation, into a Bayesian
network, from which a measure of belief is derived. This value is used to select the best alternative for further investiga-
tions. After a short theoretical excursion to Bayesian networks their use for the judgement of semantic nets is described.
4.1 Theory of Bayesian Networks
Bayesian networks are directed acyclic graphs where each node represents a random variable and the edges in between
are quantified by conditional probabilities. The structure of a Bayesian network encodes the dependency relations be-
tween the variables in the network. As the edges are established through causal relations pointing from cause to effect,
the network provides an intuitive tool to model multiple interdependencies. Bayesian networks have become popular over
the last years because it is not only possible to reason from measurements in a bottom-up fashion towards the most likely
interpretation of the observed data, but also top-down from a hypothesis towards measurements to be expected. The
theory of Bayesian networks is described in detail in (Pearl, 1988).
Each node of a Bayesian network models a discrete random variable with a finite number of different values. Initially
the belief of each node is assumed to be equally distributed, i.e. each value of the underlying random variable is given
the same probability. As soon as evidence is introduced into the net, for example by a certain observation in the data, the
belief of the corresponding random variable changes. The probability of the observed value becomes 1 while the probabil-
ities of the other values are reduced to 0. According to the causal dependencies the beliefs of related nodes are influenced,
too. The evidence is propagated through the whole network according to a dedicated algorithm distinguishing messages
from inferior and superior nodes. This propagation process is also known as belief update. The belief BEL(x) of a node
X is given by Eq. (5), where A(x) denotes the diagnostic support from the m child nodes and (x) the causal support from
the n parent nodes (Fig. 4a). The term a normalizes the vector, so that the sum of the components, i.e. the probabilities
of the individual values, becomes 1. Starting at the leaf nodes of the net the belief and subsequently the 4— and z-messages
sent to the neighbouring nodes are computed recursively until an equilibrium is reached. Special techniques exist to cope
with loops in the Bayesian network, for example the method of conditioning (Pearl, 1988).
BEL(x) = a » À(x) * a(x)
Ax) = [1 Ay (x) with : Y; child nodes of X (5)
j
a(x) = > | ota [154 | with : u; parent nodes of X
Uses
In Figure 4b a propagation for a simple Bayesian network is illustrated. The evidence introduced in node E is distributed
bottom-up and top-down until all nodes are up-to-date. An increasing number of observations reduces the ignorance
represented by equally distributed probabilities. Finally the focussed node values are used for classification or decision
making.
Figure 4: a) Diagnostic and causal support in Bayesian networks. b) Propagation of evidence introduced in node E.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 347
