Bernal S D_2010.pdf - University of Plymouth

3.3. DEHNJTION AND MATHEMATICAL FORMULATION
However, the most interesting effect happens at the mxJe Moon, where its belief shows a re­
duction with respect to the previous scenario (0.524 vs. 0616). This is a consequence of the
increased probability of Gales, which suggests il is the cause responsible of Waves, thus explain­
ing away the Moon cause. In other words, once there is an explanation for Waves, namely Gales,
the probabilily of alternative explanations, such as Mnon, is reduced. Note the probabihly of
Moon is still relatively high, as according to the conditional probabilily table P(W\G,M), both
high-level causes can coexist, and in fact, when hoih are present, the conditional probability of
Waves is higher.
3.3.3.9 Example of belief propagation with no evidence
This example serves to illustrate how belief propagation operales when there is no evidence
available. All the resulting beliefs and ihe flow of messages are depicted in detail in Figure .1.8.
Strictly speaking Ihe resulting beliefs arc not the posterior probabilities, P{X\Q), as there is no
evidence available. Instead ihey represent the marginal probabilities of the variables when the
network is in an initial equilibrium stale before presenting any evidence. Therefore it is also
useful to compare the resulting beliefs in the network in equilibrium with those when there is
evidence, to obtain a better understanding of the effects of evidence propagation.
When all the bottom-up A mes,sages received by a node show Hal distributions (i.e. no evidence
below), inevitably all the A messages sent to its parents will also show llai distrihuiions, regard­
less of the incoming n messages. This is the case of node W in Figure 3.8. The prior probability
(or evidence) at the lop causal nodes