Comment by btilly | Hacker News Reader

Comment by btilly | Hacker News Reader

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If you are arguing THAT point, then you shouldn't have disagreed with me anywhere!

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.

1 and 3 are really the same thing. If the complaint is that the frequentist test can't tell you anything if the assumed distribution wasn't the right one (which is what's happening if you would have done something different), consider the bayesian case. There one might argue you at least still have the probability of each hypothesis given the data. But that forgets that it is only the probability of each hypothesis given the data, given that those were the only possible hypotheses. And if you admit the possibility of their being other possible hypotheses, then these probabilities don't really tell you anything either.

Anyway, not sure if we're on the same page, but thanks for the discussion.

We are clearly not on the same page. Because I think that 1 and 3 are rather different things, and you don't.

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.

Okay, my last try.

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.

If you include into the priors information about the likelihood of the next child being born, you will indeed get different absolute probabilities. But you will not get different relative probabilities unless your available priors create a correlation between birth order and the likelihood of different genders for the next child if it comes. And therefore the probability of having the next birth cancels out of Bayes' formula and you wind up with the exact same conclusions from the observed data.

You certainly DON'T wind up with anything like the factor of 8 difference that frequentist techniques will see!

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