Much can be done with mere comparisons of support, but statisticians want numerical measures of the degree to which data support hypotheses. Preceding chapters show how much can be achieved by an entirely comparative study. Now I shall argue that our earlier analysis can sometimes provide unique quantitative measures. It is not yet certain that this conclusion is correct, for it requires a new postulate. However, this is interesting in its own right, and though the resulting measures will have a fairly narrow domain, their very existence is remarkable.

The following development is novel, but in order to declare its origins I call this chapter the fiducial argument. The term is Fisher's. No branch of statistical writing is more mystifying than that which bears on what he calls the fiducial probabilities reached by the fiducial argument. Apparently the fiducial probability of an hypothesis, given some data, is the degree of trust you can place in the hypothesis if you possess only the given data. So we can at least be sure that it is close to our main concern, the degree to which data support hypotheses.

Fisher gave no general instruction for computing his fiducial probabilities. He preferred to work through a few attractive examples and then to invite his readers to perceive the underlying principles. Yet what seem to be his principles lead direct to contradiction.

Despite this bleak prospect, the positive part of the present chapter owes much to Fisher, and can even claim to be an explication of his ideas. Fortunately the consistent core of his argument is a good deal simpler than his elliptic papers have suggested. There has already been one succinct statement of it: Jeffreys was able not only to describe its logic, but also to indicate the sort of postulate Fisher must assume for his theory to work. It is a shame that Jeffreys’ insights have not been exploited until now.

Logic

If this chapter does contribute either to understanding Fisher's principles or to the sound foundation of a theory of quantitative support, it will only be through careful statement of underlying assumptions and conventions. It is by no means certain what ought to be the basic logic of quantitative support by data.