13 ms·

Using simulations to approximate probabilities

acidburnNSA 5y ago
You generally need to know the fundamental probabilities in advance to sample the random numbers properly. What Monte Carlo allows you to do is combine basic probabilities in a vastly more complex system to see the effective average probabilities of the whole.
- ZephyrBlu 5y ago
  > What Monte Carlo allows you to do is combine basic probabilities in a vastly more complex system to see the effective average probabilities of the whole Could you elaborate on this a bit? I'm having a hard time seeing why and how you would combine basic probabilities with a complex system.
  - acidburnNSA 5y ago
    Sure. I'll give an example. In nuclear engineering we know the probabilities, as functions of energy, of each nuclear reaction a neutron and nucleus can have. We measured these carefully in labs. They're fundamental. Now if you load up a Monte Carlo simulation of an entire nuclear reactor and use Monte Carlo to send neutrons through you can compute the average reaction rates all around the reactor with its complex structures just with those fundamentals.
    - ZephyrBlu 5y ago
      So in this case it's complex because of the the number of particles in the system, but because you can model each one with known probabilities you can therefore simulate the full system?
      - acidburnNSA 5y ago
        Right. Lots of particles, and lots of complex geometric interfaces with abrupt changes in material properties.
  - kqr 5y ago
    Problems solved with Monte Carlo methods are often ones where you need to draw random values from oddly-shaped multidimensional distributions. The naive method of doing this is just to cover the space evenly and weight each value by its probability. This is often computationally inefficient because you spend lots of time generating samples that don't actually contribute much to the result. This is why we have methods like Metropolis--Hastings or Gibbs: these take an appropriately encoded description of the desired distribution and, through clever means, generate samples from it relatively cheaply, in proportion to how likely they are. This part of the problem is likely what the GP alluded to.

You generally need to know the fundamental probabilities in advance to sample the random numbers properly. What Monte Carlo allows you to do is combine basic probabilities in a vastly more complex system to see the effective average probabilities of the whole.

ZephyrBlu 5y ago
> What Monte Carlo allows you to do is combine basic probabilities in a vastly more complex system to see the effective average probabilities of the whole Could you elaborate on this a bit? I'm having a hard time seeing why and how you would combine basic probabilities with a complex system.
- acidburnNSA 5y ago
  Sure. I'll give an example. In nuclear engineering we know the probabilities, as functions of energy, of each nuclear reaction a neutron and nucleus can have. We measured these carefully in labs. They're fundamental. Now if you load up a Monte Carlo simulation of an entire nuclear reactor and use Monte Carlo to send neutrons through you can compute the average reaction rates all around the reactor with its complex structures just with those fundamentals.
  - ZephyrBlu 5y ago
    So in this case it's complex because of the the number of particles in the system, but because you can model each one with known probabilities you can therefore simulate the full system?
    - acidburnNSA 5y ago
      Right. Lots of particles, and lots of complex geometric interfaces with abrupt changes in material properties.
- kqr 5y ago
  Problems solved with Monte Carlo methods are often ones where you need to draw random values from oddly-shaped multidimensional distributions. The naive method of doing this is just to cover the space evenly and weight each value by its probability. This is often computationally inefficient because you spend lots of time generating samples that don't actually contribute much to the result. This is why we have methods like Metropolis--Hastings or Gibbs: these take an appropriately encoded description of the desired distribution and, through clever means, generate samples from it relatively cheaply, in proportion to how likely they are. This part of the problem is likely what the GP alluded to.