Sampling or selection bias is often presented as something that has to be overcome, avoided, or at least appropriately considered because it's a problem otherwise. I wonder how often situations arise (for statisticians) that some biased sampling is on the contrary required (to get desired results).

I find myself in the situation that I have - abstractly speaking - a bag of balls of two equally distributed colors. If I pick a ball uniformly at random it is a red or a blue ball with equal probability $p_{\textsf{red}} = p_{\textsf{blue}} = 1/2$. But then I want to get a red ball with probability $p_{\textsf{red}} = 2/3$ – for whatever reasons. And I wonder how this is best achieved.

Two methods come to my mind:

  1. Create another bag with two copies of each red ball in the original set and one copy of each blue ball. Now the probability of picking a red ball uniformly at random is $p_{\textsf{red}} = 2/3$.(1)

  2. Pick a ball uniformly at random from the original set. If it is a red one (which happens with probability $1/2$), keep it. If it is a blue one, create a random number $r$. If $r > \theta$, put it back and pick another ball uniformly at random. Keep it.

How has $\theta$ to be chosen to end up with a red ball with probability $2/3$? Assume $\theta = 0$, i.e. when you pick a blue ball you always put it back and pick another ball. With probability $1/2$ it is a blue ball again, and you end up with this. So the probability that you end up with a red ball is $p_{\textsf{red}} = 1/2 + 1/2\cdot 1/2 = 3/4$?(2) To achieve $p_{\textsf{red}} = 2/3$ you have to choose $\theta$ such that $1/2 + 1/2\cdot (1-\theta)\cdot 1/2 = 2/3$, i.e. $\theta = 1/3$.

My questions are:

  • Is this procedure sound?

  • Are there possible pitfalls?

  • Is it possibly a common procedure to efficiently perform biased sampling?

  • How is it named?

Finally: What are typical situations - from the point of view of a statistician or "stochastician" - where one needs this kind of biased sampling? Some specific (or abstract) use cases would be welcome!

Background: My specific use case is from stochastic network generation:

When applying the standard configuration model to generate graphs with a given degree sequence, the resulting graphs have almost no clustering (i.e. a very small mean clustering coeffient). To avoid this and to generate graphs with a given mean clustering coefficient, you have to choose pairs of stubs to be connected not uniformly at random, but have to favour those which result in triangles (triadic closure). So having picked one stub, among those to choose next the ones with distance 3 from the first one are to be chosen with higher probability. (This approach could be an alternative to M.J.E. Newman's generalization of the standard configuration model with prescribed triangle distribution.)

It may be not by chance that my use case is from the context of stochastic synthesis and not from statistical analysis where biased sampling always is to be avoided. But I may be wrong.

(1) If you want to pick several samples without replacement, then after picking a red copy you have to remove the other copy, too.

(2) If you want a higher probability than $3/4$ of getting a red ball, you have to put back blue balls at least a second time. If you want to get a red ball with probability $1$ (as long as there is a red ball), you have to put back blue balls until you get a red one, which eventually will happen. This is equivalent to searching for red balls and pick them.

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    $\begingroup$ Research the Radon-Nikodym Theorem and information about "change of measure." The Girsanov Theorem could be a good place to begin. $\endgroup$ – whuber May 18 at 13:53
  • $\begingroup$ @whuber: There seems to be a long way from my simple question to the rather sophisticated concept of "change of measurw $\endgroup$ – Hans-Peter Stricker May 19 at 15:44
  • $\begingroup$ ... measure". Can you help me to make the first steps from the one to the other? $\endgroup$ – Hans-Peter Stricker May 19 at 15:45
  • $\begingroup$ You appear to want to change the probabilities in a stochastic process: that's what "change of measure" means. $\endgroup$ – whuber May 19 at 15:50
  • $\begingroup$ @whuber: My question is: why do others need and perform a "change of measure"? When is it justified? What shall be achieved by it? I can answer these questions for my own use case, but what are other use cases? $\endgroup$ – Hans-Peter Stricker May 20 at 8:55

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