Bernal S D_2010.pdf - University of Plymouth

3.3. DEFINITION AND MJOUEMATICAL FORMULATION
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Figure 3.9: Example nt'belief propagation inairec-siruciured network. The network has three
levels iirganizeil in a tree slructare. In the first step evidence propagates from
two of [he child nodes in the lower level, leading lo the updiite of ihe belief in
Ihe iniermediale nodes, in the second step, the helief at the lop level is updated,
together with Ihebelief at the lower-levcrchiJdnodes thai hadn't been instantiated.
The crucial process occurs in slep ihree when a message is sent downward from
the lop node. Noie this didn't hjp|>en in ihe network of the previous examples,
where the propagation ended once the message reached the top nixies. The reason
is Iha! in ihis case ihe lop node receives messages from the two inlennediale child
nodes (the left and [he righi branches of ihe Iree), and therefore ii musi generaie a
top-down message for each node conveying the evidence collected from Ihe other
node. In oihcr words the evidence from the left branch must be propagated to the
nodes in ihe right branch and vice versa. This is shown in steps three and tour.
updating the belief of Fallen trees. Analogously, evidence originating from Ihe node Fallen
trees would update the belief of all the nodes in Ihe opposite branch, by flowing up to node
Gales and down through node Waves lo nodes Surfing and Fishing.
We can therefore distinguish between three types of networks. The first two fall into the cat­
egory of singly-connected networks, those without loops, and the number of steps required to
obtain the exact beliefs of all nodes is proportional lo the diameter of Ihe network. In singly-
connected tree networks (no loops wiiJi one parent per node), evidence propagates from the leaf
nodes to the root node and back down again (Figure 3.9), This happens because outgoing A
messages do no! lake into account the n me,ssage received from the parent node. In singly-
connected polytrees (no loops with multiple parents), there is a single bottom-up top-down
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