Barnard et al’s Foundations of Stats. Inference
G.A. Barnard and D.R. Cox (Eds.) The Foundations of Statistical Inference: A Discussion, Methuen, 1962.
This provides a good discussion between the main proponents of different views on the subject.
Barnard
… it is not true that we can ever enumerate all possible hypotheses. …
Savage
… I certainly agree that the danger of losing serendipity by binding oneself to an over-rigid model is pone against which we cannot be too alert. We must not pretend to have enumerated all the hypotheses in some simple and artificial enumeration that actual excludes some of them. The list can however be completed … by adding a general ‘something else’ hypothesis and this will be quite workable, provided that you can tell yourself in good faith that ‘something else’ is rather improbable.
Barnard
… What one requires … is not just that there should be some hypotheses, but that they should enable you to compute probabilities for the data, and that requires very well defined hypotheses.
Lindley
All probabilities are conditional [on the set of hypotheses considered].
Barnard
I agree.
Pearson
I suggest that you start by knowing perfectly well that they are conditional and when you come to the answer you forget about it.
Barnard
[You] can only compare probabilities on different sets of evidence if those probabilities are conditional on the same set of assumptions. …
Savage
[We] may not even think of some important relevant hypotheses … . These imperfections in the theory of personal probability are real and render its conclusions imperfect. We must, therefore, use the theory circumspectly, checking it frequently with common sense. … Not knowing what to conclude is a reality not to be escaped by adopting any so-called “exact” theory or rule.
As an aside:
… If the client had sufficient time, energy and talent he could be his own statistician, and it seems to me that the first object for theoretical study is such a statistically well endowed investigator. …
Comments
The above aside seems to me an important one. My experience is that analysts often take the attitude that their technical training gives them the right to over-ride whatever insights the client may have, whereas I feel sure that some clients would not accept these views no matter how well these clients had been trained in the technical area. (For example, they would not accept that certain commonplace technical assumptions are actual true of their case at hand.) But the aspiration should be that the result of the client and the technical adviser (e.g., statistician) getting together should be the same as if the client had all the necessary technical understanding.
Savage’s position is very reasonable, much more so than that commonly credited to him, for example that in every situation you should make your best probability guestimates and then act as if they were well-founded.
djmarsay 