There are several ways to clarify this so-called fiducial argument. One way employs a so-called functional model, i.e., the specification of a statistical model by means of a particular function. For the above model, the function is  \[ f(\theta, \epsilon) = \theta + \epsilon = \hat{\theta}(s) . \]  It relates possible parameter values \(\theta\) to a quantity based on the sample, in this case the estimator of the observations \(\hat{\theta}\). The two are related through a stochastic component \(\epsilon\) whose distribution is known, and the same for all the samples under consideration. In our case \(\epsilon\) is distributed normally with variance \(1 / \sqrt{n}\). Importantly, the distribution of \(\epsilon\) is the same for every value of \(\theta\). The interpretation of the function \(f\) may now be apparent. Relative to the choice of a value of \(\theta\), which then obtains the role of the true value \(\theta^{\star}\), the distribution over \(\epsilon\) dictates the distribution over the estimator function \(\hat{\theta}(s)\).