popper-logic-scientific-discovery

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new appendices
statistical interpretations may be used as excellent and weighty
test-statements.* 4
Thus our analysis shows that statistical methods are essentially
hypothetical-deductive, and that they proceed by the elimination of
inadequate hypotheses—as do all other methods of science.
10. If δ is very small, and therefore also P(e)—which is possible
only for large samples—then we have, in view of (6),
(7)
P(e, h) ≈ P(e, h) − P(e).
In this case, and only in this case, it will therefore be possible to
accept Fisher’s likelihood function as an adequate measure of degree
of corroboration. We can interpret, vice versa, our measure of degree of
corroboration as a generalization of Fisher’s likelihood function; a generalization
which covers cases, such as a comparatively large δ, in which Fisher’s
likelihood function would become clearly inadequate. For the likelihood
of h in the light of the statistical evidence e should certainly not
reach a value close to its maximum merely because (or partly because)
the available statistical evidence e was lacking in precision.
It is unsatisfactory, not to say paradoxical, that statistical evidence e,
based upon a million tosses and δ = 0.00135, may result in numerically
* 4 This point is of considerable interest in connection with the problem of the numerical
value of the absolute probabilities needed for the determination of C(x, y), i.e. the
problem discussed under point 3 of the ‘Second Note’, and also in the present note (see
especially footnote *1). Had we to determine the absolute probability of the ‘total
available evidence’ consisting of the conjunction of a large number of observational
reports, then we should have to know the absolute probability (or ‘width’) of each of
these reports, in order to form their product, under the assumption (discussed in appendix
*vii above) of the absolute independence of these reports. But in order to determine
the absolute probability of a statistical abstract, we do not have to make any assumptions
concerning either the absolute probability of the observational reports or their
independence. For it is clear, even without assuming a Laplacean distribution, that (6)
must hold for small values of δ, simply because the content of e must be always a measure
of its precision (cf. section 36), and thus absolute probability must be measured by the width
of e, which is 2δ. A Laplacean distribution, then, may be accepted merely as the simplest
equiprobability assumption leading to (6). It may be mentioned, in this context, that the
Laplacean distribution may be said to be based upon a universe of samples, rather than a
universe of things or events. The universe of samples chosen depends, of course, upon
the hypothesis to be tested. It is within each universe of samples that an assumption of
equiprobability leads to a Laplacean (or ‘rectangular’) distribution.