A new look at fiducial inference

Since the idea of fiducial inference was put forward by Fisher, researchers have been attempting to place it within a rigorous and well motivated framework. It is fair to say that a general definition has remained elusive. In this paper we start with a representation of Bayesian posterior distributions provided by Doob that relies on martingales. This is explicit in defining how a true parameter value should depend on a random sample and hence an approach to "inverse probability" (Fisher, 1930). Taking this as our cue, we introduce a definition of fiducial inference that extends existing ones due to Hannig.