Salsburg, D. 2001.
``
The Lady Tasting Tea
-- How statistics revolutionized science
in the twentieth century''.
Owl Book.
Chapter 12. The Confidence Trick
Neymans's solution
How does one comupte an interval estimate?
How does one interpret an interval estimate?
Can we make a probability statement about an interval
estimate?
How sure are we that true value of the parameter lies
withing the interval?
In 1934, Jerzy Neyman presented a talk before the Royal
Statistical Society, entitled ``On the Two Different
Aspects of the Representative Method.'' His paper dealt
with the analysis of sample surveys; it has the elegance
of most of his work, deriving what appear to be simple
mathematical expressions that are intuitively obvious
(only after Neyman has derived them). The most important
part of this paper is in an appendix, in which Neyman
proposes a straightforward way to create an interval
esitmate and to determine how accurate that estimate is.
Neyman called this new procedure ``confidence intervals,''
and the ends of the confidence intervals he called
``confidence bounds.''
R.A. Fisher was also among the discussants, but he missed
this point. His discussion was a rambling and confused
collection of references to thing that Neyman did not
even include in his paper. This is because Fisher was in
the midst of confusion over the calculation of interval
estimates. In his comments, he referred to ``fiducial
probability,'' a phrase that does not appear in Neyman's
paper. Fisher had long been struggling with this very
problem -- how to determine the degree of uncertainty
associated with an interval estimate of a parameter.
Fisher was working at the problem from a complicated angle
somewhat related to his likelihood function. As he quickly
proved, this way of looking at the formula did not meet
the requirements of a probability distribution. Fisher
called this function a ``fiducial distribution,'' but then
he violated his own insights by applying the same
mathematics one might apply to a proper probability
distribution. The result, Fisher hoped, would be a set of
values that was reasonable for the parameter, in the face
of the observed data.
This was exactly what Neyman produced, and if the
parameter was the mean of the normal distribution, both
methods produced the same answers. From this, Fisher
concluded that Neyman had stolen his idea of fiducial
distribution and given it a different name. Fisher never
got far with his fiducial distributions, because the
method broke down with other more complicated parameters,
like the standard deviation. Neyman's method works with
any type of parameter. Fisher never appeared to understand
the difference between the two approaches, insisting to
the end of his life that Neyman's confidence intervals
were, at most, a generalization of his fiducial intervals.
He was sure that Neyman's apparent generalization would
break down when faced with a sufficiently complicated
problem -- just as his own fiducial intervals had.
