Yes, that’s correct. z = f(\text{random_variable1, random_variable2, ...}), with no error. If the values of the random variables are known, then the exact value of z is known.
Thanks for the suggestion, I’ve tried to use Dirac before and yes, the lack of proper sampling is an issue. Regarding z ~ Normal(f(SomeRandomStuff), epsilon), are you suggesting I should remove the error term \epsilon from the equation y_t = k * x_t + \epsilon, and instead add it into the z_t equation z_t = \text{Normal}(\text{lag_transform}(\lambda, y_t), \,\epsilon_\text{new})? This could potentially work, however knowing the actual distribution of the original \epsilon is desirable.
Otherwise, if you meant to add an extra \epsilon term to the model, I’m not sure how this would be helpful. Is the idea to give it a very small prior, so that now the sampling works as intended and the added \epsilon has a negligible impact on the estimation of the other parameters?