If I have a deterministic, analytic model, $y=f(x)$, I can analytically calculate the uncertainty in $y$ from a known uncertainty in $x$, $\sigma$. Or I can do a Monte Carlo integration: sample from the distribution $x$, and run those samples through the model, and get an estimate of the distribution of $y$ from the model's output.
Is there any guarantee that the results of the two methods will be (asymptotically) the same? If so, is it equivalent for a larger class of models than just those with analytical solutions?
Best Answer
I think the question is to which order you calculate, the analytic method.
The usual, Gaussian error propagation does a first-order approximation of the function around a certain point and varies with the deviation. If the function has a lot of higher-order components (in that region) or you are dealing with comparable large uncertainties, you will end up with bad estimates.
MC on the other side will give you usually the "best" prediction but use up way more resources. Although you can account for higher order approximations in your error propagation, you basically do not know where to "stop" in order to get enough orders.
For the limit, AFAIK, they yield the same result (in the limit of infinite order and infinite MC runs). A simple reasoning: the errors are intrinsically given (even dough we don't know them). Both methods try to approximate those as good as possible and in the limit yielding the correct errors.