**Note on an Article by Sir Ronald Fisher **

*By Jerzy Neyman (1956)
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**Summary**

**(1) FISHER’S allegation that, contrary to some passages in the introduction and on the cover of the book by Wald, this book does not really deal with experimental design is unfounded. In actual fact, the book is permeated with problems of experimentation. (2) Without consideration of hypotheses alternative to the one under test and without the study of probabilities of the two kinds, no purely probabilistic theory of tests is possible. (3) The conceptual fallacy of the notion of fiducial distribution rests upon the lack of recognition that valid probability statements about random variables usually cease to be valid if the random variables are replaced by their particular values. The notorious multitude of “paradoxes” of fiducial theory is a consequence of this oversight. (4) The idea of a “cost function for faulty judgments” appears to be due to Laplace, followed by Gauss.**

*1. Introduction*

In a recent article (Fisher, 1955), Sir Ronald Fisher delivered an attack on a a substantial part of the research workers in mathematical statistics. My name is mentioned more frequently than any other and is accompanied by the more expressive invectives. Of the scientific questions raised by Fisher many were sufficiently discussed before (Neyman and Pearson, 1933; Neyman, 1937; Neyman, 1952). In the present note only the following points will be considered: (i) Fisher’s attack on the concept of errors of the second kind; (ii) Fisher’s reference to my objections to fiducial probability; (iii) Fisher’s reference to the origin of the concept of loss function and, before all, (iv) Fisher’s attack on Abraham Wald.

THIS SHORT (5 page) NOTE IS NEYMAN’S PORTION OF WHAT I CALL THE “TRIAD”. LET ME POINT YOU TO THE TOP HALF OF p. 291, AND THE DISCUSSION OF FIDUCIAL INFERENCE ON p. 292 HERE.

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Thanks Mayo, a very interesting response that I had not read before. I was particularly taken by this sentence:

“The relation between the two schools of thought might be compared to that between tactics (Fisher) and strategy (Wald).”

I like it because it seems to be a nice alternative to what I think of as being a focus on local probabilities (the probability of observing data as extreme as _this_ data given _this_ null hypothesis which allows tactical inference about this experiment) and global probabilities (the probability of observing data this extreme using this method, which allows strategic decisions on the basis of long-run test performance). Of course, the distinction comes from different choices of conditioning, and thus the different reference sets that Fisher and Neyman preferred.

Neyman’s objections to the notion of fiducial probability are interesting in light of the above because his insistence that a confidence interval with numerical bounds does not tell anything about the value of the particular parameter of interest is an insistence on a strategic interpretation over a tactical interpretation. It may be difficult to hold both interpretations at the same time, but the fact that I prefer the tactical should not mean that you should not prefer the strategic.

Michael:The probabilistic instantiation alleged to follow deductive by Fisher here is false, fallacious. The only surprising thing is that Neyman didn’t bring out the error more powerfully here. Neyman never said you learn nothing about values of parameters via CIs, he never would have developed them had he thought so. The conclusion is the interval estimate. The warrant comes from the error probs of the estimator–and it’s not mere long run coverage..