next up previous
Next: The geometry of belief Up: Adjusting beliefs in stages Previous: Path correlation

Adjustment in several stages

Now suppose that we intend to adjust B sequentially by the collections of quantities tex2html_wrap_inline4712 . We define the cumulative collection

displaymath4339

and denote the cumulative adjustment

displaymath4695

We may `partial out' any stage of the adjustment, defining for any tex2html_wrap_inline4718 the partial adjustment of B by tex2html_wrap_inline4369 given tex2html_wrap_inline4371 as

displaymath4696

Corresponding to the adjustment tex2html_wrap_inline4728 is the bearing tex2html_wrap_inline4730 . The bearing for the partial adjustment tex2html_wrap_inline4732 is

displaymath4697

As before, we have

displaymath4698

The bearing for the partial adjustment expresses the change in both magnitude and direction in beliefs between stages [j] and [i]. The tex2html_wrap_inline4403 stepwise partial adjustment, tex2html_wrap_inline4728 is

displaymath4699

with bearing tex2html_wrap_inline4754 . We refer to the full sequence of stepwise adjusted bearings tex2html_wrap_inline4756 as the data trajectory. For each j we may write

displaymath4700

We have therefore have that

displaymath4701

where

displaymath4702

So to examine the ways in which the individual terms combine to determine the revision we must consider



David Wooff
Thu Oct 15 11:56:54 BST 1998