Given some (unnormalized) probability distribution function \(p(\vec{x})\), we can estimate \(f(\vec{x})\)
$$ \langle f \rangle = \frac{\int p(\vec{x})f(\vec{x})d^Dx}{\int p(\vec{x}) d^Dx} $$
Using stochastic methods to evaluate the above integral is more effiecient than non-stochastic integration in high dimensions
- D>4 for trapezoid
- D>8 for Simpson
- D>2\(\alpha \) generally
However, we’ll run into trouble if
- The variance of the function scales with the number of dimensions
- If the probability function is hard to sample (a delta function for example)
- The probability function is negative
The high dimensional functions that we care about are usually wave functions, which is close to zero in the majority of its phase space due to physical reasons. Therefore a trapezoid type of method will waste the majority of its time sampling points where the function is close to zero.