Bernal S D_2010.pdf - University of Plymouth

3,3. i^^lNITION AND MATHEMATICAL FORMVLATION
dition established by converging nodes, described in Section 3.3.2, and matches our intuition
regarding multiple causes. Without any information about the state of the Waves, data on the
slate of Gales should not influence the state of Moon, as they are conditionally independent
causes.
3.3.3.10 Example of belief propagation with tree structure
The previous examples are all based on a network with the same simple structure; a central node
with two parents and two children node. Although this type of structure serves to demonstrate
the main concepts behind belief propagation, it does not capture an inten2,sting effect of h^e-
stnictured networks, which is therefore described in this subsection.
In this case the network has three levels organized in a tree structure as shown in Figure 3.9. In
the first step, evidence propagates from two of the child nodes in the lower level, leading to the
update of the belief in the intcnnediaie nodes. In the second step, the belief at the top level is
updated, together with the belief of the lower-lever child nodes that hadn't been instantiated.
The crucial process occurs in step three when a message is sent downward from the top node.
Note this didn't happen in the network of the previous examples, where Ihe propagation ended
once the message reached the top nodes. The reason is that in this case the top node receives
messages from the two intermediate child nodes (the left and the right branches of the tree), and
therefore it must generate a top-down message for each node convcyinp the evidence collected
from the other node. In other words the evidence from the left branch must be propagated to the
nodes in the right branch and vice versa. This is depicted graphically in steps three and four.
In the original example, an equivalent flow of evidence would happen, for example, if node
Gales had a second child node, such as Fallen frees. Evidence originating in the node Surfing
would propagate up the node Waves to the root node Gales and back down the opposite branch.
'Caption for Figure 3.8. Cjtample of belief propagation wiih no evidence. When all the boltomup A messages
received by a node show Hat ilistribuiions, as is ihe erase of node W, jnevKabiy all [he A messages sent to its parent
nodes will also show flat distributions, regardless of the incoming n messages. The prior probability (or evidence)
at Ihe top causal nodes 0 and M does not influence the other causal node, until their common child W gathers
some diagnostic evidence. This reflects the d-separntion condiLioii established by converging nodes, described in
Section 3.,^.2, and matches our intuition regarding multiple causes. Without iiny inlormaiiiin about the state of the
Woven, data on the state of Gnies should not influence the stale of Moon, as they are conditionally independent
causes,
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