The functional relation f(theta, epsilon) is smoothly invertible, such that f^{-1}(theta_hat(s), epsilon) = theta_hat(s) - epsilon = theta maps each combination of theta_hat(s) and epsilon to a unique parameter value theta.

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