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The size and bearing of the adjustment

 

The bearing is a quantity which summarises the actual effects of adjustment. Its principal properties are that

The bearing for this adjustment is the linear combination

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or, standardising the tex2html_wrap_inline8938 's,

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We noted above the property that the changes from prior expectation to adjusted expectation for quantities in tex2html_wrap_inline8606 are equivalent to the covariances of the quantities with the bearing. For example, for tex2html_wrap_inline8580 ,

eqnarray941

can be seen to be the difference between its expectation tex2html_wrap_inline9188 and its adjusted expectation tex2html_wrap_inline9190 . In this way, changes in expectation are expressible solely through a covariance with the bearing, and so the magnitude of a change in expectation for any quantity depends only upon the strength of correlation between the quantity and the bearing, and upon the length (i.e. the variance) of the bearing.

The variance of the bearing is a quantity that we refer to as the size of the adjustment. Here it is

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We define the expected size of the adjustment to be the prior expectation (where we replace tex2html_wrap_inline9196 by tex2html_wrap_inline9198 ) for the size of the adjustment. For our example, it is

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We can show that the size of the adjustment always turns out to be equal to the resolved uncertainty for the belief structure, tex2html_wrap_inline9202 , as seen in section 4.3. In this sense we have that the expected magnitude of change in expectation is equal to the expected reduction in uncertainty over the entire base.

The ratio of the size of the adjustment to its expected size is termed the size ratio. Its value here is

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We interpret the three size statistics as follows. The size of the adjustment (here equal to 0.3179) summarises the magnitude of actual change over the entire belief structure tex2html_wrap_inline8554 . Relative to its prior variance, every linear combination in tex2html_wrap_inline8610 has an actual squared change in expectation no greater than this value. For example, we can be sure that the actual change in expectation in the quantity tex2html_wrap_inline9214 is bounded according to

displaymath9155

where tex2html_wrap_inline9222 . (In fact the actual change in expectation turns out to be rather close to this bound, being tex2html_wrap_inline9224 .) We contrast the value for the size of the adjustment to its expectation by calculating their ratio, the size ratio. For our example the value is 0.9389, very close to its expectation of unity. Generally, the magnitude of the size ratio serves as a further diagnostic test: a size ratio distant from unity may cause us to reconsider our specifications, or to check our data values for spurious entries, and so forth.


next up previous
Next: Adjusting beliefs in stages Up: Observing the adjustment Previous: Consistency checks for the

David Wooff
Thu Oct 15 12:20:04 BST 1998