What annoys me with Monte Carlo methods is trying to get self-consistent statistics at multiple parameter values. E.g., in your example, what is the derivative

What annoys me with Monte Carlo methods is trying to get self-consistent statistics at multiple parameter values. E.g., in your example, what is the derivative of the solid angle as the detector is moved? Or more generally, if we have some 'net goodness' measure for a system depending on parameters, how can we efficiently maximize it, when simulations are noisy and basins are shallow? My understanding is that these sorts of questions come up in ML, and there are ways of dealing with it, but they can't converge nearly as fast as simple iterations like Newton's method. Even if I have to take a series approximation instead of a simple formula, I'll be able to use autodiff (or at worst, symbolic differentiation) to get quick and precise answers to these questions.