If two sampling functions and are used to estimate the value of , then our Monte Carlo estimate becomes:
where is the number of samples taken from a distribution method , and is the number of samples taken from a distribution method .
and are special weighting functions chosen such that the value of this estimator is the value of the integral above.
Balance heuristic
Using the balance heuristic as and , we get:
Adam also had this on the board, for some reason that I’ll probably discover eventually (it feels important):
Power heuristic
The power heuristic modifies the balance heuristic by squaring each term in and , which further reduces the variance or something. So use that instead.