But if they are having children until they have one of each, the probability of the different observations does depend on the prior events!
If the set of prior theories does not include anything that depends on birth order, then birth order and the experimental design are irrelevant to the posterior conclusions.
We have 3 types of facts.
1. What is the set of prior theories? Probability theory says this should matter. Bayesians are explicit about its involvement. Frequentists ignore it.
2. Observed data. Probability theory says that this should matter. Everyone takes this into account.
3. Experimental design for what would have happened had something different than the observed actually happened. This matters a lot to frequentist approaches and does not matter at all to Bayes' theorem. Bayesian approaches generally do not care about it at all.
The difference between scenario 1 and scenario 2 is a fact of type 3, the conditions under which Bill and Lorena would have stopped having children. Unless you believe that Bill and Lorena's desire for one gender has a material impact on the probability of boys vs girls, this fact is irrelevant to any calculation of posterior probabilities. And is irrelevant in classical Bayesian approaches.
Yet, despite being irrelevant, it matters a lot for frequentist approaches.
Anyway, not sure if we're on the same page, but thanks for the discussion.
In particular 1 consists of exact statements about the likelihood of 7 births in a row being mmmmmmf.
By contrast 3 consists of statements about what Bill and Lorena's childbearing plans would have been if something different had happened.
Those are very different types of statement. There is no connection between statements of type 3 and statements of type 1 unless Bill and Lorena's state of mind makes a significant difference to the odds of the next child being a boy.
For Bayes' theorem, we need a theory of how the data is produced given the parameter of interest. Bill and Lorena's plans certainly influence what data I observe: in scenario two, I can never observe the data BBBBGGG, but in scenario one, I can. My point is that your first category is not "exact statements about the likelihood of 7 births in a row being mmmmmmmf", it is "exact statements about the likelihood of observing mmmmmmf", which is, in fact, quite different, if you admit the possibility of Bill and Lorena having particular childbearing plans.
You certainly DON'T wind up with anything like the factor of 8 difference that frequentist techniques will see!