  The fiducial is not unique in general, but we prove that in a restricted
class of models it is uniquely determined by the sampling distribution of the
data. It depends in particular not on the choice of a data generating model.
The arguments lead to a generalization of the classical formula found by Fisher
(1930). The restricted class includes cases with discrete distributions, the
case of the shape parameter in the Gamma distribution, and also the case of the
correlation coefficient in a bivariate Gaussian model. One of the examples can
also be used in a pedagogical context to demonstrate possible difficulties with
likelihood-, Bayesian-, and bootstrap-inference. Examples that demonstrate
non-uniqueness are also presented. It is explained that they can be seen as
cases with restrictions on the parameter space. Motivated by this the concept
of a conditional fiducial model is introduced. This class of models includes
the common case of iid samples from a one-parameter model investigated by
Hannig (2013), the structural group models investigated by Fraser (1968), and
also certain models discussed by Fisher (1973) in his final writing on the
subject.
