A) I’m talking to someone and they tell me they have two children. “Oh, do you have any boys?” I ask, “I love boys!”. They nod.
B) I’m talking to someone and they tell me they have two children. One of the children then runs up to the parent. It’s a boy.
The chance of two boys is clearly 1⁄3 in the first scenario, and a half in the second.
The scenario in the question as asked is almost impossible to answer. Nobody would ever state “I have two children, at least one of whom is a boy.” in real life, so there’s no way to update in that situation. We have no way to generate good priors. Instead people make up a scenario that sounds similar but is more realistic, and because everyone does that differently they’ll all have different answers.
In scenario B, where a random child runs up, I wonder if a non-Bayesian might prefer that you just eliminate (girl, girl) and say that the probability of two boys is 1/3?
In Puzzle 1 in my post, the non-Bayesian has an interpretation that’s still plausibly reasonable, but in your scenario B it seems like they’d be clowning themselves to take that approach.
So I think we’re on the same page that whenever things get real/practical/bigger-picture, then you gotta be Bayesian.
I don’t really see how? A frequentist would just run this a few times and see that the outcome is 1⁄2.
In practice, for obvious reasons, frequentists and bayesians always agree on the probability of anything that can be measured experimentally. I think the disagreements are more philosophical about when it’s appropriate to apply probability to something at all, though I can hardly claim to be an expert in non-bayesian epistemology.
I agree that frequentists are flexible about their approach to try to get the right answer. But I think your version of the problem highlights how flexible they have to be i.e. mental gymnastics, compared to just explicitly being Bayesian all along.
Consider two realistic scenarios:
A) I’m talking to someone and they tell me they have two children. “Oh, do you have any boys?” I ask, “I love boys!”. They nod.
B) I’m talking to someone and they tell me they have two children. One of the children then runs up to the parent. It’s a boy.
The chance of two boys is clearly 1⁄3 in the first scenario, and a half in the second.
The scenario in the question as asked is almost impossible to answer. Nobody would ever state “I have two children, at least one of whom is a boy.” in real life, so there’s no way to update in that situation. We have no way to generate good priors. Instead people make up a scenario that sounds similar but is more realistic, and because everyone does that differently they’ll all have different answers.
In scenario B, where a random child runs up, I wonder if a non-Bayesian might prefer that you just eliminate (girl, girl) and say that the probability of two boys is 1/3?
In Puzzle 1 in my post, the non-Bayesian has an interpretation that’s still plausibly reasonable, but in your scenario B it seems like they’d be clowning themselves to take that approach.
So I think we’re on the same page that whenever things get real/practical/bigger-picture, then you gotta be Bayesian.
I don’t really see how? A frequentist would just run this a few times and see that the outcome is 1⁄2.
In practice, for obvious reasons, frequentists and bayesians always agree on the probability of anything that can be measured experimentally. I think the disagreements are more philosophical about when it’s appropriate to apply probability to something at all, though I can hardly claim to be an expert in non-bayesian epistemology.
I agree that frequentists are flexible about their approach to try to get the right answer. But I think your version of the problem highlights how flexible they have to be i.e. mental gymnastics, compared to just explicitly being Bayesian all along.