## A Pseudo-Natural Example

While it may seem that we should be able to test whether CVE is
worthwhile for natural examples by comparing it to VE for standard
examples, it isn't obvious that this is meaningful. With the
table-based representations, there is a huge overhead for adding a new
parent to a variable, however there is no overhead for making a
complex function for how a variable depends on its existing
parents. Thus, without the availability of effective algorithms that
exploit contextual independence where there is a small overhead for
adding a variable to restricted contexts, it is arguable that builders
of models will tend to be reluctant to add variables, but will tend to
overfit the function for how a variable depends on its parents. As all
models are approximations it makes sense to consider approximations
to standard models. As we are not testing the approximations
[10][26], we will pretend these are the real models.
In this section we produce evidence that there exists networks for
which CVE is better than VE. The sole purpose of this experiment it to
demonstrate that there potentially are problems where it is
worthwhile using CVE. We use an instance of the *water* network [17]
from the Bayesian network repository^{8}
where we approximated the conditional probabilities to create
contextual independencies. Full details of how the examples were
constructed are in Appendix *. We collapsed probabilities that were within 0.05 of each other to create
confactors. The *water* network has 32 variables and the tabular
representation has a table size of *11018* (after removing variables
from tables that made a difference of less that 0.05). The contextual
belief network representation we constructed had 41 confactors
and a total table size of *5834*.

Scatterplot of runtimes (in msecs) of CVE (x-axis) and VE
(y-axis) for the

*water* network. Full details are in
Appendix

*.

Figure * shows a scatter plot of 60 runs of random
queries^{9}. There were 20 runs each for 0, 5 and 10 observed
variables. The raw data is presented in Appendix *. The
first thing to notice is that, as the number of observations
increases, inference becomes much faster. CVE was often significantly
faster than VE. There are a few cases where CVE
was much worse than VE; essentially, given the elimination ordering,
the context-specific independence didn't save us anything in these example, but we had
to pay the overhead of having variables in the context.

David Poole
and Nevin Lianwen
Zhang,Exploiting Contextual
Independence In Probabilistic Inference, Journal of
Artificial Intelligence Research, 18, 2003, 263-313.