Fiducial prediction

A real-valued fiducial process is a process in the probabilistic sense, but the probability distributions are not defined on Borel sets. The adjective ‘fiducial’ indicates that a group acts on the observation spaces and the probability distributions are defined on the smaller σ-algebra of group-invariant Borel sets. Since a different group action determines a different σ-algebra, the group is an integral part of a fiducial process. A fiducial process obeys the rules of probability theory, but care is required in the calculation of conditional distributions, such as are required for purposes of prediction and Bayesian inference. For a simple fiducial model, the one-step prediction is an ordered pair (ĝ, P ) consisting of a group element ĝ together with the conditional distribution P of (ĝ, Yn+1) on the invariant σ-algebra on G×R. More conventionally, the fiducial prediction is an ordered pair consisting of a reference point or estimator ĝ together with the conditional distribution of the pivotal statistic ĝ−1Yn+1 on (R,B) given the σ-algebra generated by the observation. A fiducial model with parameter space Θ is a set of processes in which the group also acts on the parameter space, and all probability distributions are defined on the groupinvariant σ-algebra. Several examples of fiducial models are described, mainly with the aim of understanding the nature and limitation of fiducial predictions. A Bayesian analysis is described for fiducial models, and it is shown that if the group acts transitively on the parameter space, the prior has no effect on the conclusion. The Bayesian conclusion does not emerge immediately in the form of a probability distribution on the parameter space. Instead it occurs in the form of an ordered pair (ĝ, P ) consisting of a reference point ĝ ∈ G together with the conditional distribution P of (ĝ, θ) on G-invariant sets in G×Θ. The Bayesian conclusion may be re-expressed in the form of a conditional confidence density on the parameter space, in agreement with Fisher’s (1934) analysis for location and location-scale models. This derivation is entirely probabilistic, without recourse to improper distributions.