Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

and α is a normalizing constant, determined by
α = ∑v ∈(true, false) P(ev - /V). P(V/ev + ) (9.14)
It seems from the last expression that v could assume only two values:
true and false. It is just an illustrative notation. In fact, v can have a number of
possible values.
Let node V have n offsprings, vide fig. 9.5. For computing λ(V), we
divide ev - into n disjoint subsets eZ i, 1≤ i ≤n, where Zi is a child of V.
So, λ(V) = P(ev - /V).
= P(eZ1 - , eZ2 - , …, eZn - / V)
= P(eZ1 - /V). P(eZ2 - /V)…. P(eZn - /V).
= Π n i=1λ Zi.(V) (9.15)
U
V
λZ1.(V) λV (U)
… .
Z1 Z2 Z3 Zn
Fig. 9.5: Propagation of λs from the children to the parent
in an illustrative tree.
We now compute ∏(V) using the message ∏V (U) = P(U|ev + ) from the parent
U of V.
∏(V) = P(U|ev + )
= ∑u ∈ (true, false) P ( V | eV + , U = u) P(U = u |ev + )
= ∑u ∈ (true, false) P(V |U = u). P(U = u | ev + )
= ∑u∈ (true, false) P(V| U = u). ∏ v(U = u)
= [P(V|U)] T 2 x 2 x [ ∏ v (0) ∏ v (1)] T 2 x1 (9.16)