note: This question was inspired by Assessing the accuracy of a deterministic mathematical model. I have tried to give a more explicit example and specific question.
A meteorological model that predicts the weather is deterministic, so for any set of inputs it will give the same output. Commonly, a weather forecast will use today's observed meteorological conditions to predict tomorrow's high temperature. We will call the current conditions $x$, the model $g$, and the estimate of tomorrow's high temperature $\hat{T}$:
$\text{T}_\text{model}=g(x)$
Tomorrow, I observe that the high temperature was $\text{T}_\text{obs}$, with an uncertainty due to an observation error of $\epsilon_\text{obs}\sim \text{N}(0,\sigma)$.
There is no estimate of model uncertainty - theoretically, it could be as low as $\pm0.0001$ or as high as $\pm\infty$, but given the way that the model is used with fixed inputs, the model can only make a discrete estimate of a continuous variable.
Is it possible to say that the forecast is correct? Specifically, can I test the hypothesis that $\text{T}_\text{model}=\textrm{T}_\text{obs}$?
Perhaps $\text{T}_\text{model}$ falls inside the 95%CI for $\text{T}_\text{obs}$, but since the 95%CI for $\text{T}_\text{model}$ could be >> the 95%CI for $\text{T}_\text{obs}$, then it isn't clear that the hypothesis can be tested. So, can model performance be evaluated without an estimate of model uncertainty, or is an estimate of model uncertainty required?