## fiducial inference In connection with my recent tale of the many ε’s, I received from Gunnar Taraldsen [from Tronheim, Norge] a paper [jointly written with Bo Lindqvist and just appeared on-line in JSPI] on conditional fiducial models.

“The role of the prior and the statistical model in Bayesian analysis is replaced by the use of the fiducial model x=R(θ,ε) in fiducial inference. The fiducial is obtained in this case without a prior distribution for the parameter.”

Reading this paper after addressing the X validated question made me understood better the fundamental wrongness of fiducial analysis! If I may herein object to Fisher himself… Indeed, when writing x=R(θ,ε), as the representation of the [observed] random variable x as a deterministic transform of a parameter θ and of an [unobserved] random factor ε, the two random variables x and ε are based on the same random preimage ω, i.e., x=x(ω) and ε=ε(ω). Observing x hence sets a massive constraint on the preimage ω and on the conditional distribution of ε=ε(ω). When the fiducial inference incorporates another level of randomness via an independent random variable ε’ and inverts x=R(θ,ε’) into θ=θ(x,ε’), assuming there is only one solution to the inversion, it modifies the nature of the underlying σ-algebra into something that is incompatible with the original model. Because of this sudden duplication of the random variates. While the inversion of this equation x=R(θ,ε’) gives an idea of the possible values of θ when ε varies according to its [prior] distribution, it does not account for the connection between x and ε. And does not turn the original parameter into a random variable with an implicit prior distribution.

As to conditional fiducial distributions, they are defined by inversion of x=R(θ,ε), under a certain constraint on θ, like C(θ)=0, which immediately raises a Pavlovian reaction in me, namely that since the curve C(θ)=0 has measure zero under the original fiducial distribution, how can this conditional solution be uniquely or at all defined. Or to avoid the Borel paradox mentioned in the paper. If I get the meaning of the authors in this section, the resulting fiducial distribution will actually depend on the choice of σ-algebra governing the projection.

“A further advantage of the fiducial approach in the case of a simple fiducial model is that independent samples are produced directly from independent sampling from [the fiducial distribution]. Bayesian simulations most often come as dependent samples from a Markov chain.”

This side argument in “favour” of the fiducial approach is most curious as it brings into the picture computational aspects that do not have any reason to be there. (The core of the paper is concerned with the unicity of the fiducial distribution in some univariate settings. Not with computational issues.)

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