Simple formulas only work because the models themselves for those polls are incredibly simple and adding a bit more complexity requires a lot of tools to compute these uncertainties (this is part of the reason you see probabilistic programming so popular for people doing non-trivial polling work).
There are no simple approximations for a range of even slightly complex models. Even some nice computational tricks like the Laplace approximation don't work on models with high numbers of parameters (since you need to compute the diagonal of the Hessian).
A good overview of the situation is covered in Efron & Hastie's "Computer Age Statistical Inference".
There are no simple approximations for a range of even slightly complex models. Even some nice computational tricks like the Laplace approximation don't work on models with high numbers of parameters (since you need to compute the diagonal of the Hessian).
A good overview of the situation is covered in Efron & Hastie's "Computer Age Statistical Inference".