The fiducial method and invariance

1930. In papers since that time Fisher has frequently discussed aspects of the method and may have, in the view of some readers. modified or altered his ideas concerning the underlying principles and the more general aspects of the method. The method has been frequently criticized adversely in the literature, particularly in recent years. Some of the grounds for this criticism are: conflict with the confidence method in certain problems; non-uniqueness in certain problems; disagreement with 'repeated sampling' frequency interpretations; and a lack of a seemingly-proper relationship with a priori distributions. In his recent book, Statistical Methods and Scientific Inference, Fisher (1956) has devoted considerable space to the fiducial method. He states that an essential ingredient for its use is the absence of prior information concerning the value of the parameter being estimated; in his words, 'it is essential to introduce the absence of knowledge a priori as a distinctive datum in order to demonstrate completely the applicability of the fiducial method of reasoning to the particular real and experimental cases for which it was developed.' An interpretation of one aspect of this requirement might be that all parameter values are equivalent in the way in which the frequency distribution of the observable variable is related to the parameter value determining that distribution. In ? 5 this interpretation is formalized and shown to imply a mathematical model in which the parameter is related to a group of transformations on the sample space. In ?? 2 and 3 the mathematical model involving transformations is presented on its own merits and in ?? 4 and 5 the fiducial argument for it is developed. A consequence for this model is that the information about the parameter from an observed value of the variable is in the form of a frequency distribution, the fiducial distribution, having a frequency interpretation in terms of a well-defined kind of repeated sampling. This is in agreement with Fisher's statement-'the fiducial argument uses the observations (only) to change the logical status of the parameter from one in which nothing is known of it, and no probability statement can be made, to the status of a random variable having a well-defined distribution.' Another consequence in the special framework is that there is no need to require the absence of an a priori distribution for the parameter. For, if the fiducial distribution is combined in a logical manner with the a priori distribution the result is the aposteriori distribution of a Bayesian argument-a reassuring result. This is demonstrated in ? 9. A further consequence concerns prior information that the parameter value is restricted to some specified range. This restriction can be used to condition the fiducial distribution yielding a conditioned fiducial distribution. A probability combination of such restrictions can in effect generate an a priori distribution and an appropriate combination of conditioned fiducial distributions yields the Bayesian a posteriori distribution. This is discussed in ? 10.