weighting of estimates

[jim@hydra.maths.unsw.EDU.AU]

bernasch at forwiss.tu-muenchen.de (Jost Bernasch) writes:


>James Franklin writes:
 >> If you have a fairly accurate and a fairly inaccurate way of estimating
 >>something, it is obviously not good to take their simple average (that
 >>is, half of one plus half of the other). The correct weighting of the
 >>estimates is in inverse proportion to their variances (that is, keep
 >>closer to the more accurate one).
>
>Of course this is the correct weighting. Since the 60s this is done
>very succesfully with the well-known "Kalman Filter". In this theory
>the optimal combination of knowledge sources is described and
>proofed in detail.

Well, yes, in a way, but that's something like saying that the
motion of your body can be derived from Einstein's equations of
General Relativity. Too complicated. In particular, Kalman filters,
and control theory generally, are about time-varying entities, and
Kalman filters are an (essentially Bayesian) way of successively
updating estimates of a (possibly time-varying) quantity
(See R.J. Meinhold & N.D. Singpurwalla, `Understanding the
Kalman filter', American Statistician 37 (1983): 123).

The situation I was considering, and what is relevant to committees,
is much simpler (hence more general): how to combine estimates
(possibly correlated) of a single unknown quantity.

James Franklin
Mathematics
University of New South Wales