# How to continue sampling after a conditional rejection?

I encountered the following (to me) weird problem while trying to do some simple sampling.

• I have a system that generates random numbers. Let's for simplicity assume the numbers are 0, 1 and 2 with uniform probability.
• I generated one sample and that obtained sample didn't look random to me. Let's just say it was a 0 (random vector in the real system).
• Because I had some doubts about the randomness of the sample, I decided to test the system a few times (by sampling) to verify that the first sample just looks non-random to me. Turns out it is random.
• Now I want to continue sampling. This is where I'm not sure whether I keep my sample from the beginning (the 0) or if I just start over.
• If I start over isn't it like rejecting a sample conditionally on it's value (which would change the overall probability distribution)? But then again after a certain outcome was received, the chance of getting that outcome is the same as it was before.

Should I reject the first draw or not?

I hope this is not a philosophical issue in the end.

• Outcomes are (supposed to be) independent. You don't change the future distribution by rejecting $n$ samples at the beginning. You change the distribution when your process selectively reject samples and accepts the others. It's ok to restart from scratch, and it's ok to keep the first samples. – Jean-Claude Arbaut Dec 6 '18 at 14:19
• @Jean-ClaudeArbaut, thanks. I had a similar realization before but then I rejected it again because I thought that the system triggered me to reject that sample (based on the sample's value). As long as it doesn't continue triggering me to reject samples conditionally I'm fine. That's what I'm understanding. (I feel like a puppet :) – Jonathan Dec 6 '18 at 14:32