Bernal S D_2010.pdf - University of Plymouth

3.3. DEPINITION AND MATHFMATICAL FORMULATION
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Figure 3.3: Bayesian networks can have loops but not cycles, a) A cycle: a palh ihai siarls and
ends al the same node, h) A liicip: a chain with no cycles, where al leasi one node
is visited mure than once.
and ends al the same node (Figure 3.3a). A loop, however, is a chain where al least one node is
visiiec! more than once (i.e. has two or more parents), but has no cycles (Figure 3.3b),
The distinction is important because Bayesian networks by definiiion have no cycles (acyclic),
but can have loops. Bayesian networks with no loop.s are usually called singly-connected net­
works, while those with loops are called multiply-connected networks.
3.3.3 Belief propagation
3.3.3.1 Inference
Given the structure of the network and the conditional probabilities delining the joint probability
distribution (Equalion (3,9)), il is possible to analyiically tompuie the marginal probability of
each node, in terms of sums over all the possible stales of all other niKies in the system i.e.
using marginalization, as shown in Hquation (3.4). Ibr example, the marginal probability of the
variable Wean be calculated from the joint probability given by Equation 3.10 as follows.
S F (J M
As can be seen, this computation is impractical, specially for large networks, as the number of
terms in the sums grows exponentially with the number of variables, l-urthermore, there are
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