We have defined - partly functionally and partly explicitly - the model,
the quantities of interest, and the belief specifications that we need
to carry out the adjustment. In particular, we have organised matters so
that variances and covariances pertaining to underlying mean components
are stored in variance-covariance store 2, and those pertaining to the
mean-plus-residual components are in store 1. It remains only to
define a range of values for 
(we choose initially the three
values 
), and for each such value to
construct the corresponding 
, and then perform the belief
adjustments desired. The fragment of code in Figure 8 contains
three parts: calls to the main analysis subroutine with different values
of 
, the subroutine itself, and
the data. We concentrate only on describing the subroutine.
For each value of , the INDEX: and COBUILD:
commands are used to construct thirteen 
and thirteen
quantities. These will have expectations, variances, and
underlying mean-component variances computed from their definitions: the
former functionally, and the latter as linear combinations of
intercept, slope, and error quantities. (Much computation is involved at
this point, so the program may take some time to generate the
quantities, depending on computer platform.)
Once constructed, the regression parameters 
are
collected into the base named 
, and the observables are gathered
into the base whose name is 
. Observations from three experiments
are then attached: these are shown at the foot of the listing.
For the analysis, we exploit the notion of Bayes linear
sufficiency via exchangeability. Here the vector of averages of the
observables, 
, over the three
experiments is Bayes linear sufficient for the regression parameters. To
perform the Bayes linear adjustment of the regression parameters by the
observables, it is necessary to set various belief source
controls. These indicate to [B/D] the belief stores in which the overall
and underlying mean-component variance matrices are stored:
: belief
store 2. These constitute the prior variance matrix for the regression
parameters.
: belief
store 1. These constitute the overall variance matrix for the
observables.
of the observables: belief
store 2.
: belief
store 2. These constitute the covariance matrix between the regression
parameters and the observables.
The exchangeable and usedata controls specify that [B/D] should take into account data on the observables, and should use the internal routines which exploit exchangeability.
We next make two kinds of adjustments. Firstly, we adjust the regression
parameters by the data (three observations over the ), and
display their adjusted expectations using the SHOW: command. (As
the observables are Bayes linear sufficient for the regression
parameters, [B/D] automatically obtains the general adjustment via the
sample means only.) The adjusted expectations for the regression
parameters are shown in Table 3 for the three values of the
stability parameter . This output indicates that although the
model and observations are consistent with changes in expectation for
these parameters (the adjusted expectation for the intercept rises
slightly, etc.), changing the stability parameter makes little
difference, so that the model is not particularly sensitive to choice of
for predicting individual 
values.
Our second adjustment assesses variance sensitivities in
relation to changes in sample size.
It is a property of such second-order exchangeable
adjustments that the analysis for a sample of size 
can be deduced
with almost no additional computation from the same analysis performed
for a sample of size 
. The analysis is particularly simple for
adjustments where the observables are Bayes linear sufficient for the
collection to be adjusted, as is the case here. Consequently, we tend to
make an initial analysis assuming a sample of size one from which we may
deduce easily the analysis for a general sample size n.
Therefore, we use the usedata and obs controls to indicate
that the analysis should assume an initial sample size of one and that
it should ignore the actual observations available, and we then perform the
adjustment of the underlying mean component vector 
by the
observables, exploiting Bayes linear sufficiency automatically. [B/D]
deduces here that by Y we mean from the settings of the belief
source controls described above.
Much of the output available from an adjustment is available
interactively as further input to the program. These include adjusted
expectations, variances and covariances; the resolution transform and
its canonical structure, and so forth. Various other functions are
available to exploit exchangeability as appropriate. For the sensitivity
study here, we output, for each value of , the following:
function, the sample size required to
guarantee a resolution of at least 50% over the collection 
considered jointly (see [6] for a description of uncertainty
measures over collections).
For the analysis of sensitivity over the model we examine both the
canonical resolutions and some implications of changing the sample size.
For three values of , Table 4 shows two sample sizes
and thirteen canonical resolutions. The first sample size, 
, is the sample size needed to achieve a 50% reduction in
uncertainty over the collection overall, as measured by the trace of the
resolution transform. The second, 
, is the sample
size needed to guarantee a variance reduction at least 50% in
every linear combination of the quantities 
. The
canonical resolutions indicate the effective dimension of the model and
the speed of variance reduction as the sample sizes increase.
We discover that these values are highly sensitive to the choice of
stability parameter . For 
the smallest
canonical resolution is 
, so that for a sample
size n=1 we can guarantee a reduction in uncertainty of only 0.1% in
every linear combination of the mean components. This guaranteed
reduction rises to 50% if we take a sample size n=1001, whereas we
need take only n=79 to achieve the same reduction when we choose
. For uncertainty in the collection overall the picture is
similar. Examining the canonical resolutions, for 
, the
model is dominated by two canonical quantities with resolutions of 0.60
and 0.31 respectively. The remaining canonical quantities have small
resolutions, so that large sample sizes will be needed to reduce their
variances appreciably. Therefore the learning process for the model with
is essentially two-dimensional. (This should not surprise
us: taking 
forces (4) to become a simple
regression model with two parameters, intercept and common slope.)
As we reduce the stability parameter, the dynamics of adjustment
change: we increase the number of canonical quantities having noticeable
variance reductions for small sample sizes, and we learn more quickly
about the collection overall, so that for 
we can learn about
almost all combinations of the Y values. Therefore, the model is very
sensitive to choice of , for learning about changes in Y over
time.
