Multiple importance sampling

If two sampling functions $p_f$ and $p_g$ are used to estimate the value of $\int f(x)g(x)dx$, then our Monte Carlo estimate becomes:

$$\frac{1}{n_f}\sum _{i=1}^{n_f}\frac{f(x_i)g(x_i)w_f(x_i)}{p_f(x_i)}+\frac{1}{n_g}\sum _{j=1}^{n_g}\frac{f(y_j)g(y_j)w_g(y_j)}{p_g(y_j)}$$

where $n_f$ is the number of samples taken from a distribution method $p_f$, and $n_g$ is the number of samples taken from a distribution method $p_g$.

$w_f$ and $w_g$ 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 $w_f$ and $w_g$, we get:

$$w_f(x_i)=\frac{n_f{p}_f(x_i)}{n_f{p}_f(x_i)+n_g{p}_g(x_i)}\text{ and }{w}_g(y_j)=\frac{n_g{p}_g(y_j)}{n_f{p}_f(y_j)+n_g{p}_g(y_j)}$$

Adam also had this on the board, for some reason that I’ll probably discover eventually (it feels important):

$$\frac{f({\omega }_{if})\cdot \text{Li}({\omega }_{if})}{p_f({\omega }_{if})+p_l({\omega }_{if})}+\frac{f({\omega }_{il})\cdot \text{Li}({\omega }_{il})}{p_f({\omega }_{il})+p_l({\omega }_{il})}$$

Power heuristic

The power heuristic modifies the balance heuristic by squaring each term in $w_f$ and $w_g$, which further reduces the variance or something. So use that instead.