Influence diagrams contain arcs connecting nodes. We label the arcs as follows. We must first determine what kind of arcs, if any, are to be drawn. This is described in §11.4.1, with the different arc styles summarised in Table 11.4.
We will consider the case where the most detailed standard arcs might be drawn,
corresponding to arc style number 7, and so we establish a scenario
which might produce such details. The autoarc control can be
used to draw particular arc types automatically between the nodes on an
influence diagram, but we will concentrate on tailoring arcs to suit.
Suppose that we adjust 
, and produce an influence
diagram with arcs connecting the information sources nodes D, E, and F
with the node B.
We define the information flows from D, E, and F singly to B as
, 
and 
respectively, representing the
worth of each information source in the absence of any other. The
overall resolution at the node B is 
,
representing the total information arriving at B. If we observe
then diagnostic quantities for these resolutions
are given by the corresponding size ratios. Thus, for the arc between
node D and node B, 
is the ratio given by dividing the
observed maximal squared change in adjustment, 
, by its
expectation, .
We measure the information flow ``from B to D'' by the actual loss
in resolution at B if node F is withdrawn from the adjustment. In
our notation, this is 
. We measure the information
flow from B to E and from B to D similarly, by 
and 
respectively. Diagnostic quantities are
also available for these measures; the diagnostic quantity corresponding
to is 
.
An assessment of how observed information sources work together (or
against each other) is given by the path correlation which results when
one information source is extracted. Hence, we can label the arc from
node D to node B by 
(B), and the other nodes
similarly.
The arc styles consist of (1) a straight line connecting two nodes; (2)
a box containing information which is placed over the arc; and (3) a
circle drawn towards the end of the arc, containing information about
the path correlation. We describe the path correlation style first. A
path correlation lies in 
. We shade the circle according to
the absolute magnitude of the correlation, so that correlations of 1 and
-1 result in a fully shaded circle. We shade differently, according to
direction. For positive correlations, we shade a corresponding
proportion of the circle anti-clockwise, starting from 0 degrees.
For negative correlations, we shade a corresponding
proportion of the circle clockwise, starting from 0 degrees.
The size of the circle can be changed using the pcradius
control.
The box of labelling information consists of a bar divided into two. The half bar nearest the information source concerns information flow from the information source to the node being adjusted, i.e. the contribution of the information source taken singly. The half bar nearest the node being adjusted concerns information flow away from the node being adjusted to the information source, i.e. the loss from extracting the information source singly.
Consider the arc drawn from D to B, where there is no data so that
only influences are shown: Table 11.1 shows the contents of
an arc label in this case. The region
represents all the uncertainty in the
destination node, proportionately 1. The region 
represents the
uncertainty remaining after all information
sources have been fitted, 
. The region
thus represents the proprtion of uncertainty removed
in the destination node B. The region 
represents the
resolution in uncertainty in B due solely to fitting D, i.e.
. Thus, when is large and 
is small, the
implication is that the single source of information D is virtually
sufficient. When is small and is large, the
implication instead is that the single source of information D is virtually
useless. The outer half of the label is similarly configured:
is identical to , and 
is
identical to . However, the region 
represents the resolution in uncertainty in B due solely to extracting
D from the adjustment, i.e. . Thus, when
is large and 
is small, the implication is that much of the resolution in uncertainty
at 
is lost if the single source of information D is withdrawn.
When is small and is
large, the implication instead is that the single source of information
D contributes little extra to the information supplied already by
.
Now consider the case where we observe , so that we
can contrast actual to expected behaviour. The label that we draw is the
same, except that we make a further partition of the regions
and in Table 11.1. We obtain a
label which resembles that shown in Table 11.2. We partition
the region according to the magnitude of the size ratio
. If this ratio is bigger than 1, we shade the portion
, and otherwise we shade the portion
. The actual proportions of shaded
depend on whether or not the ratio is larger than 1. If it is, the
proportion of shaded is 
,
where 
by default, but can be changed using the
bigshade control. If the ratio is smaller than 1,
the
proportion of shaded is 
,
where 
by default, but can be changed using the
smallshade control. The region is partitioned
similarly, except that the pertinent size ratio is .
The proportions shaded are as described for , so that
is shaded if this ratio is larger than one, and
is shaded otherwise.
For both types of label, the labels may be modified by removing the
unresolved proportions of uncertainty given by the and
regions. This is achieved by adding 8 to the arc style.
All other regions are then
scaled up appropriately. This is useful when the actual resolution of
uncertainty in the destination is very small, but when it is still
important to understand information flow.
It is also possible to draw labels which emphasize the diagnostic
information carried by the arc. This is achieved by adding 16 to the arc
style. The effect is to obtain labels like those shown in
Table 11.2, but with the regions , ,
, and , removed and the remaining regions
scaled up.
The length and width of arc labels are set by default, but may be changed by using the arclength and arcwidth controls.
