Bernal S D_2010.pdf - University of Plymouth

3.3. DEFINITION AND MATHEMATICAL FORMULATION
all of its parents; and %(«;) represents the support the assertion t/,- = M,- receives from the
nodes above Uj. %(»,) can also be understood as the message X receives from its parent
node Ui, which combined with the messages from all other parent nodes, and multiplied by the
appropriate values of the conditional probability function, yields the n{x) function.
Therefore, as has been demonstrated, the generic node X in Figure 3.4 can calculate its own
belief if it receives the messages V; (••:) from its children and KX{UI} from its parents. In the rest
of this section we will consider how to generate these messages, which allow the evidence lo
propagate across the network.
3.3.3.3 Bottom-up mes-sattes
Taking into account that all nodes in a Bayesian networit perform the same operations, we will
consider the generic message Aj^(«,}, which node X must send lo its parent node IJ, (Figure 3,4).
It is therefore convenient to treat all parents of X, except the one receiving the message, as a
common set V, such that V —V-U, = {f/i,....(7,_i,t/n.|,...,iy/i/} as shown in Figure 3.5.
The message Xx{ui) must lake into account all evidence under die link Uj —> X, which includes
evidence coming from all other parent-s of X (Sy^ — U *(/ ;f)'- and evidence arriving from
k-\..N\i '
the descendcnts of X (e^). Given thatX separates eyj^ from ejj^ and V separates e,^;^ from U„
we can write,
[conditioning on .V and v] = ^Y,fi^vx^^x\"