Note that this command is under construction. The purpose of belief comparison diagrams is to summarise relationships
between alternative variance specifications and prediction error for a
collection of quantities.
We have in
mind that there are two alternative information sources, 
and
, with possibly different specifications about and between 
and
on the one hand, and about and between and on the
other. We will treat the two scenarios as model 1 and model 2
respectively.
Assume that the two alternative variance specifications given for
collections 
and 
are respectively 
and 
. Initially we will be concerned
with 
, but we allow the possibility
. We also restrict ourselves to the case where
data is available on and , so that we can calculate
both possible adjusted expectations for , and that we also have
available the observed value b of .
Under model 1, suppose now that the collection is adjusted by
collection . This gives rise to an adjusted version
which is some vector of linear combinations
of the elements in . This quantity is observed to be
, the vector of prediction errors for model
1. We now calculate 
and
to be its variance matrix under model 1
and model 2 respectively.
We now carry out a comparison of these two alternative variance
specifications for the adjusted version of . Let
and 
be the smallest
and largest canonical quantities (generalised eigenvalues) for this
comparison, corresponding to canonical quantities (generalised
eigenvectors) 
whose observed magnitudes are
and 
respectively. The eigenvectors are
uncorrelated under both models. The eigenvalues
, whereas the 
's are
normed to have variance unity (or, pathologically, zero) under model 1:
=1. If model 1 is correct, we should expect to see
about equal to 1. Observing close to
suggests that the data give more support to model
2. Observing distant from both unity and
suggests that the data support neither model.
The upper semicircle of each node on the comparison diagram summarises
these features. The top left quadrant is, from the centre outwards,
shaded (white,red) in the proportions
. The observed value
is plotted according to the current value of the
cdscaler control, which treats the radius in the top left
quadrant as being
equal to (by default) three standard deviations relative to model 2.
Values of larger than three standard deviations are
plotted on the quadrant boundary to indicate support for neither model.
The top left quadrant is plotted only if 
.
A large amount of red indicates that model 2 has much more variance for
some linear combinations of than does model 1.
The top right quadrant is, from the centre outwards,
shaded (white,blue) in the proportions
. The observed value
is plotted according to the current value of the
cdscaler control, which treats the radius in this
quadrant as being
equal to (by default) three standard deviations relative to model 1
(i.e. 3, as the standard deviation is unity under model 1).
Values of larger than three are
plotted on the quadrant boundary to indicate support for neither model.
The quadrant is plotted only if 
.
A large amount of blue indicates that model 2 has much less variance for
some linear combinations of than does model 1.
We repeat the adjustment process under model 2, arriving at the vector
, the observed vector
, and variance matrices 
and
.
We now carry out a comparison of these two alternative variance
specifications for this second adjusted version of . Let
and 
be the smallest
and largest canonical quantities (generalised eigenvalues) for this
comparison, corresponding to canonical quantities (generalised
eigenvectors) 
whose observed magnitudes are
and 
respectively. The eigenvectors are
uncorrelated under both models. The eigenvalues
, whereas the 
's are
normed to have variance unity (or, pathologically, zero) under model 2:
=1. If model 2 is correct, we should expect to see
about equal to 1. Observing close to
suggests that the data give more support to model
1. Observing distant from both unity and
suggests that the data support neither model.
The lower semicircle is now drawn similarly to the upper semicircle.
The bottom left quadrant is, from the centre outwards,
shaded (white,blue) in the proportions
. The observed value
is plotted according to the current value of the
cdscaler control, which treats the radius in this
quadrant as being
equal to (by default) three standard deviations relative to model 1.
Values of larger than three standard deviations are
plotted on the quadrant boundary to indicate support for neither model.
This quadrant is plotted only if 
.
A large amount of blue indicates that model 1 has much more variance for
some linear combinations of than does model 2.
The bottom right quadrant is, from the centre outwards,
shaded (white,red) in the proportions
. The observed value
is plotted according to the current value of the
cdscaler control, which treats the radius in this
quadrant as being
equal to (by default) three standard deviations relative to model 2.
Values of larger than three are
plotted on the quadrant boundary to indicate support for neither model.
The quadrant is plotted only if 
.
A large amount of red indicates that model 1 has much less variance for
some linear combinations of than does model 2.
Lines are drawn to connect the observations in diagonally opposite quadrants to indicate that the quadrants convey similar kinds of information. Observations so large as to imply plotting beyond the boundary of a node are plotted on the boundary and shaded a different colour. The radius of the small circle representing the observation can be changed via the cdradius control.