The following is an example from a book I'm reading. Let there be a sequence of throws of an unfair coin, and $\theta$ be the prob. of getting head.
Imagine the observer gets: $x=(T,H,T,T,H,H,T,H,H,H)$, which could have been obtained by different sampling processes:
- throw the coin a predetermined number of times, in this case 10.
- throw the coin until we get 6 heads.
- throw the coin until we get 3 consecutive heads.
Why would a classical/frequentist observer be interested in the sampling process when doing inference about $\theta$? I understand that a frequentist is interested not just the obtained sample, but also in every other sample that could have been obtained and wasn't. What I don't get is why is that, and how would that concern be in this specific case.
Any help would be appreciated.
Edit: For Bayesian inference, it would not matter which sampling procedure was used, as long we obtained the same sample, since then we have the same likelihood. (Assuming the prior would still be the same independently of the sampling procedure)