weighting of estimates

[Jost Bernasch]

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.

See the original work

@article{Kalman:60,
        AUTHOR = {R.E. Kalman},
        TITLE = "A New Approach to Linear Filtering and Prdiction Problems.",
        VOLUME = 12,
        number = 1,
        PAGES = {35--45},
        JOURNAL = "Trans. ASME, series D, J. Basic Eng.",
        YEAR = 1960
        }

 some neural network literature concerning this subject


@Article{WatanabeTzafestas:90,
  author = 	 "Watanabe and Tzafestas",
  title = 	 "Learning Algorithms for Neural Networks with the Kalman
                  Filter",
  journal = 	 JIRS,
  year = 	 1990,
  volume = 	 3,
  number = 	 4,
  pages = 	 "305-319",
  keywords=     "kalman, neural net"
}
@string{JIRS = {Journal of Intelligent and Robotic Systems}}

and a very good and practice oriented book

@book{Gelb:74,
        AUTHOR = "A. Gelb",
        TITLE = "Applied {O}ptimal {E}stimation",
        PUBLISHER = "{M.I.T} {P}ress, {C}ambridge, {M}assachusetts",
        YEAR = "1974"
     }

 (At least, that is the correct
 >weighting if the estimates are independent: if they are correlated,
 >it is more complicated, but not much more). Proofs are easy, and included
 >in the ref below:

For proofs and extensions to non-linear filtering and correlated
weights see the control theory literature. A lot of work is already
done!


-- Jost


Jost Bernasch       	
Bavarian Research Center for Knowledge-Based Systems 
Orleansstr. 34, D-81667 Muenchen , Germany
bernasch at forwiss.tu-muenchen.de