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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.

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    $\begingroup$ Not sure I'm understanding your question. Are you asking how do you sample from the mixture of $n$ lights? If you want to sample two or more lights, you would just repeat the procedure to sample a single light (with fresh random numbers), right? Or is there something else you're asking? $\endgroup$ – Nathan Reed Feb 14 '17 at 23:18
  • $\begingroup$ Thanks Nathan, you are correct you as to do the procedure again for multiple lights. But I wonder if the above mixture density allows one to pick two or more samples from the distribution. $\endgroup$ – ali Feb 16 '17 at 12:08
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    $\begingroup$ Are you worried about anything in particular with just repeating the process multiple times? Were you concerned about sometimes choosing the same light more than once? $\endgroup$ – trichoplax Feb 19 '17 at 20:54
  • $\begingroup$ @trichoplax. No. I just want to understand mathematically how one choose multiple samples from a merge density. $\endgroup$ – ali Feb 20 '17 at 12:53

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