Bernal S D_2010.pdf - University of Plymouth

.1,3. DEHNITION AND MATHEMATICAL FORMULATiON
Note the example described is just a toy scenario which does not accurately reflect the physical
factors affecting wave generation, or the effects waves may have on surhng and fishing activity.
Any real life situation is practically impossible to capture using such a reduced number of
variables and states. However, the analogy with a real-world situation is useful to explain the
different maihematicrtl constructs in this section.
3,3.2.2 Directiunal separation and explainin|> away
Graphical models can be divided into two categories: directed and undirected. Undirected
graphical models, also called Markov random fields, have a simple definition of independence.
Two sets of nodes A and B are conditionally independent given a third set, C, if all paths
between the nodes in A and B are separated by a node in C. By contrast, directed graphical
models (Bayesian networks), have a more complicated notion of independence, which lakes
into account the directionality of the arcs. Directionality, however, has several advantages. The
most imponanl is thai causality is clearly defined, such that an arc from A -> B indicates that
A causes B. This facilitates the construction of the graph structure, and the parameter learning
process or fitting to data. Not all causal relationships captured with directed graphical models
can be represented using undirected graphical models, and vice versa (I'earl 1988, Murphy
2002).
Before describing directional separation in Bayesian networks, it is important lo define the
concept and the different types o(evidence. An evidence function that assigns a zero probability
lo all but one slate is often said to provide hard evidence; otherwise, it is said to provide soft
evidence (e.g. 90%probability of being true and 10% probability of being false). Hard evidence
on a variable X is also often referred to as instantiation of X or lo X being observed or known.
Note that, as soft evidence is a more general kind of evidence, hard evidence can be considered
a special type of soft evidence. If the disiinciion is unimportiml we will leave out the hard or
soft qualifier, and simply talk about evidence.
Due to the directionality of arcs, there are three ditTerent types of connections in Bayesian
networks:
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