According to Peter Shirely paper, one pdf(probability density function) can be defined for the union of the light sources and pick only one using a random number from this density:

p(x) = a1 * P1(x) + (1-a1) * P2(x);

This has been referred to as mixture densities. where a=[0,1] and P1 is a pdf for selecting a point on the L1 geometry(usually 1/A). If the random number is less than a1 then the total estimate is (estimated L1)/a1.

This has been extended for N: p(x) = a1 * P1(x) + a2 * P2(x) + ... + an * Pn(x)

My question is how one pick two or multiple light sources, instead of just one, from the above density and what the estimate would be in that case.

According to Peter Shirely paper, one pdf(probability density function) can be defined for the union of the light sources and pick only one using a random number from this density:

$$p(x) = \alpha_1 P_1(x) + (1-\alpha_1) P_2(x)$$

This has been referred to as mixture densities. where $\alpha_1 \in [0,1]$ and $P_1$ is a pdf for selecting a point on the $L_1$ geometry  (usually $1/A$). If the random number is less than $\alpha_1$ then the total estimate is $(\text{estimated }L_1)/\alpha_1$.

This has been extended for N: $$p(x) = \alpha_1 P_1(x) + \alpha_2 P_2(x) + \cdots + \alpha_n P_n(x)$$

My question is how one pick two or multiple light sources, instead of just one, from the above density and what the estimate would be in that case.