In case the other responses to your question are a little difficult to parse, and to answer your question a little more directly:
- Usually, you will only get analytic answers for simple questions about simple distributions.
- For more complicated problems (either because the question is complicated, or the distribution is complicated, or both), you will need to use numerical methods.
- This doesn't necessarily mean you'll need to do many simulations, as in a Monte Carlo method, although that can be a very reasonable (albeit expensive) approach.
More direct questions about certain probabilities can be answered without using a Monte Carlo method. The Fokker-Planck equation is a partial differential equation which can be solved using a variety of non-Monte Carlo approaches. The quasipotential and committor functions are interesting objects which come up in the simulation of rare events that can also be computed "directly" (i.e., without using a Monte Carlo approach). The crux of the problem is that applying standard numerical methods to the computation of these objects faces the curse of dimensionality. Finding good ways to compute these things in the high-dimensional case (or even the infinite-dimensional case) is a very hot area of research in applied mathematics. Personally, I think unless you have a very clear physical application where the mathematics map cleanly onto what you're doing, all this stuff is probably a bit of a waste of time...
Thanks for the explanation this was very helpful. You've given me a whole new list of stuff to Google. The quasipotential/comittor functions especially seem quite interesting although I'm having a bit of trouble finding good resources on them.
They are pretty advanced and pretty esoteric. They will be very difficult to get into without a solid graduate background in some of this stuff, or unless you're willing to roll up your sleeves and do some serious learning. The book "Applied Stochastic Analysis" by Weinan E, Tiejun Li, and Eric Vanden-Eijnden is probably a decent place to start. I took a look at this book a while ago, and it's probably decent enough to get a foothold on the literature in order to figure out if this stuff will be useful for you. These guys are all monsters in the field.
- Usually, you will only get analytic answers for simple questions about simple distributions.
- For more complicated problems (either because the question is complicated, or the distribution is complicated, or both), you will need to use numerical methods.
- This doesn't necessarily mean you'll need to do many simulations, as in a Monte Carlo method, although that can be a very reasonable (albeit expensive) approach.
More direct questions about certain probabilities can be answered without using a Monte Carlo method. The Fokker-Planck equation is a partial differential equation which can be solved using a variety of non-Monte Carlo approaches. The quasipotential and committor functions are interesting objects which come up in the simulation of rare events that can also be computed "directly" (i.e., without using a Monte Carlo approach). The crux of the problem is that applying standard numerical methods to the computation of these objects faces the curse of dimensionality. Finding good ways to compute these things in the high-dimensional case (or even the infinite-dimensional case) is a very hot area of research in applied mathematics. Personally, I think unless you have a very clear physical application where the mathematics map cleanly onto what you're doing, all this stuff is probably a bit of a waste of time...