Fisher’s … Natural Sciences
Fisher Mathematical Probability in the Natural Sciences 1959
“[C]onfusion and misapprehensions abound. The requirements of a correct statement of Mathematical Probability are, I believe only three:
(a) A conceptual Reference Set, which may be conceived as a population of possibilities of which an exactly known fraction possess some chosen characteristic. To the extent that it is a meaningful fraction of the whole, the set must be measurable, but it need not be measurable in all respects. …
(b) It must be possible to assert that the subject of the probability statement belongs to this Set. … Tasks of identification … belong to the scientist, and may require his full attention. … This second requirement puts our probability statement into the real world.
(c) No subset can be recognized [as] having a different probability. …
(d) It has often been imagined that probabilities a priori can be set up axiomatically [e.g. by the principle of indifference]; this has led to many inconsistencies and paradoxes, and I am forced to conclude that the process is completely bogus.”
“[R]eally important matters such as the standardization of drugs, the control of epidemics, and the precision of ballistic missiles are liable in the future to be influenced by young men now leaving these departments armed with … confused and obsolete ideas. This, in some sort, concerns us all.“
Conditions (a) and (c) seem to me reasonable and perhaps necessary for a good, widely applicable, theory. For example, condition (c) avoids some difficulties with approaches such as Savage’s. But I can envisage reasonable variations that may have their uses: I am not clear that there is a single best theory.
Condition (b) is decisive. A mathematical theory can only connect axioms to theorems. Thus, viewed as a mathematical theory, Geometry says nothing about the real world: it is up to scientists to make that link, or not. Similarly with any mathematical theory of probability. This condition seems unavoidable.
Unfortunately, the scientific method typically relies on statistics and hence seems to rely on some aspects of a theory of probability (such as the law of large numbers). This seems to introduce some circularity into the justification of the scientific method. I think that this is a genuine problem, but that Keynes and Good have a solution.
Lindley and de Finetti sometimes appear to take a contrary view, that Bayesian probability can be justified ‘mathematically’ as always being the best method for tackling any uncertainty. But – despite the sometimes heated disputes between the sides – they do admit to some issues, which they regard as concerning the application of their strong Bayesian approach, which seem to mirror Fisher’s theoretical concerns.
As a mathematician Fisher’s approach has the advantage that the mathematical theory is mathematical (and trustworthy) in the mathematical sense, whereas the (strong) Bayesian theory is only mathematical in the sense that it employs mathematical methods, and to call it a mathematical theory on these grounds risks bringing mathematics into disrepute, as in the financial crisis of 2007/8. An mathematical theory must have axioms, so we always need to check that they hold in any particular case, such as ‘the control of epidemics’.
An implication of Fisher’s view is that one ought not to use the term ‘the probability of X, conditioned on Y’ unless one is reasonably sure that there is a genuine ‘type 1’ probability distribution, dependent on Y. Thus for every Z that overlaps Y, P(X|Y,Z) = P(X|Y), at least approximately. This usage seems to avoid some of the difficulties with conventional probability theory. One might also talk about a quasi-probability, P'(X|Y) in other cases, but then one could not impute a probability P(X|y) to a member, y, of Y.
I hope to find the time to go through the various counter-examples to numeric probability, to see how they relate to Fisher, particularly (c). For example, suppose that a doctor gives you advice based on ‘gold standard’ trials, but you have contrary experience in your familily. What do you do?
Other works on probability.