Fiducial probability, recognisable sub-sets and Behrens' test

While preparing an obituary memoir on Sir Ronald Fisher for the Royal Society (Yates and Mather, [1963]) I re-read the literature on fiducial probability, particularly that on Behrens' test for the difference of the means of two samples from normal populations with unknown variances. The validity of this test has been extensively questioned, and it is still not accepted in many quarters; in consequence doubt has been cast on the whole Fisherian concept of fiducial probability. It may therefore be appropriate, in this Memorial Number, to review the arguments that have been advanced on both sides, and to try and understand exactly what Fisher had in mind. For although, as Fisher himself stated in Statistical Methods and Scientific Inference ([1956], IV, 6), 'it is difficult to find realistic data which present this problem' the logical issues are of great importance, and failure to recognise them is leading modern statistics into many false and indeed ridiculous situations, which can only bring discredit on our science.