As I understand, Monte Carlo integration uses stochastic sampling to sample points.
Obviously, you would need many samples to reach an accurate result, but why does this process have to be random?
Would using a symmetrical grid of very dense samples (e.g. a 1 million by 1 million grid) achieve the same goal?
Are there any benefits to random sampling?
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I just wanted to revisit this question to provide what I feel is a more intuitive explanation of why a randomly sampled algorithm is faster at providing a result.
Look at the two image below. Both do not show the complete, full quality image. They are both approximations of the actual image (which is analogous to the graph of some function).
You can view the blurred image as the randomly sampled one and the mostly black image as the uniformly distributed one. The same number of sample points can be used for computing both of the below images.
However, by randomly sampling those few points, we can get an idea of what we are approximating faster. The same cannot be said of the uniformly distributed image, where all the samples were "spent" showing, in full detail, only a small portion of the overall image. I believe this is why, intuitively, random sampling can give us a better result.
It doesn't have to be completely random. In fact, using semirandom sample points is an active area of research. For instance the Latin Hypercube or Sobol' sequence.
The reason it often is random is that nonrandom sample points can magnify certain biases present in the function to be estimated. And also, of course, because it's easier.