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Adjusting the canonical belief structure

 

If we wish, we can define a new base tex2html_wrap_inline8994 (where tex2html_wrap_inline8956 and tex2html_wrap_inline8966 are the canonical directions given in the preceding section) to contain our belief grid, and examine the adjustment of the corresponding belief structure tex2html_wrap_inline9000 by the belief structure tex2html_wrap_inline8806 . Amongst other results, we find that the summary of uncertainty over the belief structure tex2html_wrap_inline9000 is the same as that for the belief structure tex2html_wrap_inline8554 :

displaymath8992

This is because the elements tex2html_wrap_inline9016 constitute a basis for the vector space tex2html_wrap_inline8610 , so that the corresponding belief structures tex2html_wrap_inline8554 and tex2html_wrap_inline9000 are equivalent for this purpose. We also obtain the standardised adjusted expectation formula for each direction, approximately

eqnarray808

A further property here is that just as the original directions tex2html_wrap_inline9028 are uncorrelated a priori, so too the adjusted expectations for the canonical directions, tex2html_wrap_inline9030 are uncorrelated.



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