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Adjusting beliefs in stages

In our example so far we have adjusted one belief structure, tex2html_wrap_inline8554 , by another, tex2html_wrap_inline8806 . Have we exhausted our exploratory possibilities, or are there extra insights to be had by approaching the problem in a different way? Well, suppose that we consider the analogy of a traditional multiple regression, where we try to predict a response variable tex2html_wrap_inline8818 from a collection of regressors tex2html_wrap_inline8820 . In terms of this analogy we may be interested not only in the predictive power of the collection taken as a whole but also in whether every tex2html_wrap_inline8822 is useful for the prediction; whether certain subsets of the tex2html_wrap_inline8822 's are more useful than others; and so forth.

For our Bayes linear approach we are concerned with similar (but broader) interests, which we investigate by applying the notion of adjusting beliefs in stages. An illustration arises from queries raised about our example so far: two of our analyses to date (the standardised values for the adjusted expectations of both the original quantities and the canonical directions) suggest actual changes in expectation substantially larger than expected. Simultaneously we expect to learn very little about tex2html_wrap_inline8582 (as its resolution is only about 5%) and tex2html_wrap_inline8966 (its resolution is only some 2%), but their changes in expectation are relatively very large. Various evidence points to a surprisingly large value of tex2html_wrap_inline8768 being at fault. Suppose, then, that we consider tex2html_wrap_inline8546 and tex2html_wrap_inline8550 as being two distinct sources of information, and suppose that we adjust tex2html_wrap_inline8554 firstly by tex2html_wrap_inline8546 , and then by tex2html_wrap_inline8550 . (In what follows we use the notation tex2html_wrap_inline8806 and tex2html_wrap_inline9256 synonymously.)





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