Fiducial inference concerns inversion of estimation. Instead of predicting that from 10 flips of a fair coin 5 will show heads in the average, one concludes from the result of 6 heads among 10 flips a certain probability that the coin is fair.
The controversy about the fiducial argument has not yet been settled, as the following statements show: "The aim of fiducial probability ... seems to be what I term making the Bayesian ommelette without breaking the Bayesian eggs" (Savage, 1963). "The fiducial argument has had very limited success and is now essentially dead" (Pederson, 1978). "A few subsequent attempts have been made to resurrect fiducialism, but it now seems largely of historical importance, particularly in view of its restricted range of applicability when set alongside models of current interest" (Davison, 2001).
However some applications have been shown in Bayesian iterative simulation through Markov chain Monte Carlo runs. Tsionas (2013) has applied three approaches, namely fiducial inversion, bootstrap approximation, and structural simulation approximation. Another recent application is by Hannig (2009) on wavelet regression.
So probably Seidenfeld et al. (1992) should be given the last word: " it certainly remains a valuable addition to the statistical lore."
L.J. Savage: Discussion. Bull. Inst. Internat. Statist. 40 (1963) 925-927.
J.G. Pederson: Fiducial inference, Int. Stat. Rev. 46 (1978) 147–170.
A.C. Davison: "Biometrika Centenary: Theory and general methodology", Biometrika (2001).
E. G. Tsionas: Fiducial approximations to posterior distributions (2013).
J. Hannig, T. Lee: Generalized fiducial inference for wavelet regression Biometrika, 96(4) (2009) 847–860.
T Seidenfeld: R. A. Fisher's Fiducial Argument and Bayes' Theorem, Statistical Science, 7, No. 3. (1992) 358-368.
Further see wikipedia on fiducial inference.