  Fisher's fiducial argument is widely viewed as a failed version of Neyman's
theory of confidence limits. But Fisher's goal -- Bayesian-like probabilistic
uncertainty quantification without priors -- was more ambitious than Neyman's,
and it's not out of reach. I've recently shown that reliable, prior-free
probabilistic uncertainty quantification must be grounded in the theory of
imprecise probability, and I've put forward a possibility-theoretic solution
that achieves it. This has been met with resistance, however, in part due to
statisticians' singular focus on confidence limits. Indeed, if imprecision
isn't needed to perform confidence-limit-related tasks, then what's the point?
In this paper, for a class of practically useful models, I explain specifically
why the fiducial argument gives valid confidence limits, i.e., it's the "best
probabilistic approximation" of the possibilistic solution I recently advanced.
This sheds new light on what the fiducial argument is doing and on what's lost
in terms of reliability when imprecision is ignored and the fiducial argument
is pushed for more than just confidence limits.
