Bernal S D_2010.pdf - University of Plymouth

3.3, DEFINmON AND MATHEMATICAL FORMULAVON
P{Xl,....XN)=^l\P{X^\nx^ (3.9)
i
Note for nodes without parents (root nodes), the conditional probability of X, is equal to its
prior probabiUty. i.e. P{X,]X\x,) — P{Xi). Thus, defining the whole structure of a Bayesian net­
work requires specirtcalion of the conditional probability distribution of each node with parents,
P(A^4njv,).plus the prior probability distributions of all root nodes,/'(Xr,„„).
More formally, a Bayesian network is a pair B — {G, /'), where
• G — (V,^) is an acyclic directed graph with V = {Xi,X2,....X„}, a set of nodes (vertices):
and A =C V x V, a set of arcs defined over the nodes;
• PiV). a joint probability distribution over V. given by Equation (3.9).
An explanation of why ihc griiph is denoted as acyfl.it: and direaed, and why these two prop­
erties are imponant, can be found further down in this section after intnxlucing a clarifying
example.
3.3.2.1 .^n inu.strative example
Figure 3.2 shows a Bayesian network with six random variables representing a toy model sce­
nario which can be used to illustrate the above concepts. For simplicity we use discrete binai^y
variables, i.e. each variable can be in either of two states, true or false. However, in a real
scenario these variables are typically either continuous, or discrete with several slates.
The scenario assumes the presence of big waves in the sea is a consequence of two causes: the
presence of gales, as strong winds are associated with large wind-generated waves; and whether
the moon is aligned with the sun or not. When the moon and the sun are aligned (which occurs
during full moon and new moon periods) their gravitational force is combined increasing the
amplitude of tidal waves. The presence of big waves is represented by the variable Wave.f (W);
the presence of gaies is represented by the variable Gale.s (G); and whether the moon is aligned
with the sun or not is represented by the variable Moon (M).
Because both Gale.s and Moon have no parent nodes, they are considered to be root nodes, and
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