The idea of the fiducial argument can now be expressed succinctly. It is to project the distribution over the stochastic component back onto the possible parameter values. The key observation is that the functional relation \(f(\theta, \epsilon)\) is smoothly invertible, i.e., the function  \[ f^{-1}(\hat{\theta}(s), \epsilon) = \hat{\theta}(s) - \epsilon = \theta \]  points each combination of \(\hat{\theta}(s)\) and \(\epsilon\) to a unique parameter value \(\theta\). Hence, we can invert the claim of the previous paragraph: relative to fixing a value for \(\hat{\theta}\), the distribution over \(\epsilon\) fully determines the distribution over \(\theta\). Hence, in virtue of the inverted functional model, we can transfer the normal distribution over \(\epsilon\) to the values \(\theta\) around \(\hat{\theta}(s)\). This yields a so-called fiducial probability distribution over the parameter \(\theta\). The distribution is obtained because, conditional on the value of the estimator, the parameters and the stochastic terms become perfectly correlated. A distribution over the latter is then automatically applicable to the former (cf. Haenni et al, 52-55 and 119–122). 