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Fig. 5.4
Nonparametric bootstrap estimate for a feature
f
of a Bayesian network
An introduction to the relevant theory, applications, and related techniques (such as
the
jackknife
) is provided in the classic monograph by
Efron and Tibshirani
(
1993
).
In Bayesian networks, bootstrap is used to investigate the properties of the pa-
rameters of the network, such as in
Koller and Friedman
(
2009
), or of its structure,
such as in
Friedman et al.
(
1999a
). An illustration of the parallel implementation of
such an approach is provided in Fig.
5.4
. In both cases, the aspects being investi-
gated are usually the expected value or the variance of some aspect of the Bayesian
network. For example, in
Friedman et al.
(
1999a
) the statistics of interest were the
probabilities associated with particular structural features of the network, such as
Markov blankets or different topological orderings of the nodes. In the analysis of
the
Sachs et al.
(
2005
) data covered in Sect.
2.5
, the statistics of interest were the
probabilities associated with each arc and its directions.
There are many other features that have a practical significance in a Bayesian
network. We may be interested, for example, in the sparseness of the network we
learned from the
hailfinder
data set using the hill-climbing algorithm. Sparse
networks are particularly useful in analyzing real-world data: they are easier to inter-
pret and inference is computationally tractable. We can use
bn.boot
and
narcs
to derive a point estimate and a confidence interval for the number of arcs as follows.
> sparse = bn.boot(hailfinder, algorithm = "hc",
+ R = 200, statistic = narcs)
> summary(unlist(sparse))
Min. 1st Qu. Median
Mean 3rd Qu.
Max.
63.00
64.00
65.00
64.69
65.00
67.00
> quantile(unlist(sparse), c(0.05, 0.95))
5% 95%
64 66
hailfinder
has 56 nodes, so with 65 arcs it can be considered sparse. Further-
more, we can see that the bootstrap estimate has a very low variance; the boundaries
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