Figure: A join tree quantified with numbers (a), and with Q-DAG nodes (b).
We now show how the proposed Q-DAG algorithm can be used to generate a Q-DAG for the belief network in Figure 4(a).
Figure: A join tree quantified with numbers (a), and
with Q-DAG nodes (b).
We have only one evidence variable in this example, 
. And we are interested
in generating a Q-DAG for answering queries about variable 
, that is,
queries of the form 
.
Figure [*](a) shows the join tree for the belief network in
Figure 4(a), where the tables contain the potential functions
needed for the probabilistic clustering algorithm.
Figure [*](b) shows the join tree again, but the tables
contain the potential functions needed by the Q-DAG clustering algorithm.
Note that the tables are filled with Q-DAGs instead of numbers.
We now apply the Q-DAG algorithm. To compute the Q-DAG nodes that will evaluate
to , we must compute the posterior distribution 
over cluster
since this is a cluster to which variable belongs. We can then
sum the distribution over variable 
to obtain what we want. To compute
the distribution we must first compute the message 
from
cluster 
to cluster .
The message is computed by summing the potential function
of cluster over all possible values of variable , i.e.,
which leads to:
The posterior distribution over cluster , , is computed using
which leads to
The Q-DAG node 
for answering queries of the form is
computed by summing the posterior over variable ,
leading to
which is the Q-DAG depicted in Figure 4(b).
Therefore, the result of applying the algorithm is two Q-DAG nodes, one
will evaluate to 
and the other will evaluate to
under any instantiation 
of evidence
variables 
.