Cyan comments on Outside the Laboratory - Less Wrong
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If Alice decides to conduct 12 trials, then the sampling distribution of the data is the binomial distribution. If Alice decides to sample until 3 successes are achieved, then the sampling distribution of the data is the negative binomial distribution. These two distributions are proportional when considered as functions of the parameter p (i.e., as likelihood functions). So in this specific case, from a Bayesian point of view the sampling mechanism does not influence the conclusions. (This is in contradistinction to inference based on p-values.)
In general, you are correct to say that biased data collection is not irrelevant; this idea is given a complete treatment in Chapter 6 (or 7, I forget which) of Gelman et al.'s Bayesian Data Analyses, 2nd ed.