Review of the example

We began with quantities and , quantities representing the blood glucose levels before, and two hours after, ingestion of a certain quantity of sugar for an ``average'' or ``typical'' elderly person. To learn about these quantities, an elderly doctor takes the similar quantities and relating specifically to herself, and makes various specifications (expectations and covariances) over these quantities; and observes and .

Our approach shows that the following ``answers'' are consistent with the doctor's inputs.

• she revises her expectation for upwards from 4.16 to 4.71; and for upwards from 6.25 to 6.91. After standardising, we notice that the latter change is surprisingly large. We also notice that the adjusted expectation for is very close to a diagnosis threshhold.
• she reduces her variance for from 1.12 to 0.77, and for from 2.43 to 2.32, reductions of some 31% and 5% respectively.
• she examinines the canonical directions and discovers that she is learning mostly in only one direction, the first canonical direction , suggesting that if she wished to improve her estimates, she should concentrate on learning about the other canonical direction, , perhaps by considering further observables that she feels might be informative about the speed of reaction of an elderly person's body to various stimuli.
• the standardised adjusted expectation formula for shows that is the most influential component of . Thus she checks her data values and finds that the standardised value for is much larger than its expectation.
• by examining the bearing, she discovers the most important direction for actual adjustment, and she observes that the size ratio is 0.9389, suggesting that the influence of the data for the change in expectation was about as expected.
• some of the analyses suggest actual changes in expectation significantly larger than expected. We explored our data and specifications further by adjusting in stages, and found two points of interest:
  • the two-hour measurement is not expected to be informative in addition to what can be learned from , so that is essentially non-informative;
  • it is precisely this portion that produces surprising changes in expectation given its lack of information content.
  • these conclusions can be drawn also from withdrawing from the adjustment and examining the effects upon the inflation of uncertainties and the changes in expectation.

Without exploring the inputs further, one or both of the following interpretations appear plausible. Firstly, our Doctor may not be as typical a healthy elderly patient as she believes herself to be: this interpretation is supported by her fasting measurement also being rather larger than expected. Secondly, she may be more certain about her expectations than she should be. If she is sure that she is healthy, then her hypothesis that the elderly are misdiagnosed is supported by this analysis; however she has allowed insufficient leeway in the variance specifications for the data to be consistent with her beliefs. In a sense she is ``more right'' about her doubts about the ogt-test than she suspected.

Finally, whereas the analysis has suggested these conclusions as being consistent with the Doctor's inputs, neither the inputs nor the conclusions are definitive for various reasons. For example,

• she might have provided more detailed inputs if she had been prepared to put in the extra work necessary beforehand;
• the analysis leaves open some questions;
• the analysis suggests that some of her inputs might be untrustworthy.

Therefore, she takes the adjustments of belief provided by the analysis as inputs helpful in forming her actual revised opinions. For example, the analysis has revealed that she is learning very little about , so a natural continuation of her analysis would be to explore ways (perhaps a more detailed prior specification) of increasing her knowledge about .

On another level, several technical issues are raised. For example, having performed the analysis, her judgement about whether or not she actually believes in the results of her analysis is itself a further statement of belief, itself subject to the laws of coherence over time. Such matters, although outside the scope of this document, are discussed in [1].