Bernal S D_2010.pdf - University of Plymouth

3.3. DEFINWONAND MATHEMATICAL FORMULATION
We assume the following initial conditions; the iwo root nodes M and C lake the value of their
prior probabilities; child node F acts as an anlicipatory node, thus maintaining a flal distribution,
and node S is an evidence node with a dislrihulion (0,1), i.e. hard evidence indicating that
there is definitely Surfinf; activity, tiiven this information and ihe condilional probability tables
provided In Figure 3.6, all the nodes are ready to send messages to the intermediate node IV.
The initial conditions, and the subsequent generated messages following Equations (3.30) and
(3.31), are shown below:
'Capiion for Figure 3.6. Example of belief propaga[ii}n wilh diagnosiic evideiice. The scenario, based an the
previously described loy example, shows how evide[ice abou[ Surfing propagates across ihe ne[w(irk, updating the
beUefs in all mhcr nodes. In the ftrsi step Ihe iwo rod nodes iind the two leaf nodes send messages to ihe intermediate
node IV. which multiplies the combined lop down evidence, !t{w) and boiiom-up evidence, A(»'), toohiaiiiiis belief.
Step two shows the ouigoititi messages irimi node W lo its child and parent nodes, Ihi^ key property here is thai
ihe ouigoing messages take inio account all evidence encepi that which originated Iroiii the desiinaiiim node, f-or
example the message from W to G, Aiv(s)- doesn'l take into account the prior inlomimion conveyed through the
incoming message Kw[g]. The belief, or posterior probability, of both parent nodes is higherihan Iheir original prio/"
probabilily. Node F also u|idaies its belief according lo ihe new incoming message, showing a lower probability.
Iniuiiively. evidence of surfing activity suggests the pre.scnee of big waves, which in turn suggests ihe presence of
gales and/or ihe moon as ihe generaiing causes. Ai the same time, ihe presence of big waves suggests lishing activity
Is less likely to be present. See the text for a detailed step by step description of the mathematical operations.
95