Re: The Basis for Quadratic Taguchi Loss Function

["Jonathan Siegel" <jmsiegel@umich.edu>]

A Taylor series requires a function which is everywhere differentiable
in the area of interest. Some very important functions, including Myron
Tribus' William Tell example and Deming's missing-the-train example, are
not only not differentiable but not even continuous. Whether a person's
losses form a function sufficiently smooth for the Taylor machinery to
apply is an empirical question of fact and personal value that cannot be
derived from mathematical first principles. A person using the Taylor
series to explain matters must therefore simply assume that people's
losses have a character amenable to the formula.

Deming often pointed out that there no logical necessity for real-world
variation to conform to the mathematical laws of probability. In just
the same way, there is no logical necessity for the losses real-world
people experience to conform to the rules of the Taylor formula. The
only justification for either mathematical model is that it often seems
to work. It often happens to be so; it's a useful rule of thumb. But it
is evidence, not mathematics, that determines whether this is so. 
 
I would respectfully suggest that one may be misplacing ones faith if
one bases ones actions on the derivation of mathematical formulae. Such
formulae always implicitly make assumptions about the nature of the
world. In the absence of evidence, these assumptions are untested,
unsupported, and may prove disastrous. Moreover, when they prove false,
mathematicians seem to be just as quick to adopt theories making the new
assumptions inherent in the nature of mathematics as they were to build
theories embedding the old assumptions. Thus Riemannian geometry
displaced Euclidean geometry after the discovery of relativity, and the
logic of C.I. Lewis displaced the logic of Aristotle to explain the
physics of the quantum. 
 
Jonathan Siegel
jmsiegel@yahoo.com