Conditional on the value of the estimator, the parameters and the stochastic terms become perfectly correlated, so a distribution over the stochastic terms is automatically applicable to the parameters.

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