334 14 Bayesian Networks
not(PF) PF
not(Flu) and not(Cold) 0.99 0.01
not(Flu) and (Cold) 0.90 0.10
(Flu) and not(Cold) 0.10 0.90
(Flu) and (Cold) 0.05 0.95
The CPD for the Temperature (T) node has two incoming edges, so its CPD
will have four entries as in the case above, but because T is continuous, it
must be specified using some technique other than a table. For example, one
could model it as a normal distribution for each of the four cases as follows:
Mean Std Dev
not(Flu) and not(Cold) 37.0 0.5
not(Flu) and (Cold) 37.5 1.0
(Flu) and not(Cold) 39.0 1.5
(Flu) and (Cold) 39.2 1.6
As an example of one term of the JPD, consider the probability of the event
(Flu)and not(Cold)and(PF)and(T≤ 39 .0). This will be the product of
the four probabilities: Pr(Flu), Pr(not(Cold)) = (1-Pr(Cold)), Pr(PF|Flu and
not(Cold)), and Pr(T≤39.0|Flu and not(Cold)). Multiplying these gives
(0.0001)(.99)(.90)(0.5) = 0. 004455.
Although the BN example above has no directed cycles, it does haveundi-
rectedcycles. It is much harder to process BNs that have undirected cycles
than those that do not. Some BN tools do not allow undirected cycles be-
cause of this.
Many of the classic stochastic models are special cases of this general graph-
ical model formalism. Although this formalism goes by the name ofBayesian
network, it is a general framework for specifying JPDs, and it need not in-
volve any applications of Bayes’ law. Bayes’ law becomes important only
when one performs inference in a BN, as discussed below. Examples of the
classic models subsumed by BNs include mixture models, factor analysis,
hidden Markov models (HMMs), Kalman filters, and Ising models, to name
afew.
BNs have a number of other names. One of these,belief networks,hap-
pens to have the same initialism. BNs are also called probabilistic networks,
directed graphical models, causal networks, and “generative” models. The
last two of these names arise from the fact that the edges can be interpreted
as specifying how causes generate effects. One of the motivations for intro-
ducing BNs was to give a solid mathematical foundation for the notion of