In your example the experimenter has learned whether you have cancer. And she reflects that knowledge in the structure of the experiment: you are woken up 9 times if you have the disease.
Set aside the amnesia effects of the drug for a moment, and consider the experimental setup as a contorted way of imparting the information to the patient. Then you’d agree that with full memory, the patient would have something to update on? As soon as the second day. So there is, normally, an information flow in this setup.
What the amnesia does is selectively impair the patient’s ability to condition on available information. it does that in a way which is clearly pathological, and results in the counter-intuitive reply to the question “conditioning on a) your having woken up and b) your inability to tell what day it is, what is your credence”? We have no everyday intuitions about the inferential consequences of amnesia.
Knowing about the amnesia, we can argue that Beauty “shouldn’t” condition on being woken up. But if she does, she’ll get that strange result. If she does have cancer, she is more likely to be woken up multiple times than once, and being woken up at all does have some evidential weight.
All this, though, being merely verbal aids as I try to wrap my head around the consequences of the math. And therefore to be taken more circumspectly than the math itself.
If she does condition on being woken up, I think she still gets 1⁄2. I hate to keep repeating arguments, but what she knows when she is woken up is that she has been woken up at least once. If you just apply Bayes rule, you get 1⁄2.
If conditioning causes her to change her probability, it should do so in such a way that makes her more accurate. But as we see in the cancer problem, people with cancer give the same answer as people without.
Then you’d agree that with full memory, the patient would have something to update on?
Yes, but then we wouldn’t be talking about her credence on an awakening. We’d be talking about her credence on first waking and second waking. We’d treat them separately. With amnesia, 2 wakings are the same as 1. It’s really just one experience.
I’m not sure what more I can say without starting to repeat myself, too. All I can say at this point, having formalized my reasoning as both a Python program and an analytical table giving out the full joint distribution, is “Where did I make a mistake?”
Where’s the bug in the Python code? How do I change my joint distribution?
I like the version of your halfer variant version of your table. I still need to think about your distributions more though. I’m not sure it makes sense to have a variable ‘woken that day’ for this problem.
In your example the experimenter has learned whether you have cancer. And she reflects that knowledge in the structure of the experiment: you are woken up 9 times if you have the disease.
Set aside the amnesia effects of the drug for a moment, and consider the experimental setup as a contorted way of imparting the information to the patient. Then you’d agree that with full memory, the patient would have something to update on? As soon as the second day. So there is, normally, an information flow in this setup.
What the amnesia does is selectively impair the patient’s ability to condition on available information. it does that in a way which is clearly pathological, and results in the counter-intuitive reply to the question “conditioning on a) your having woken up and b) your inability to tell what day it is, what is your credence”? We have no everyday intuitions about the inferential consequences of amnesia.
Knowing about the amnesia, we can argue that Beauty “shouldn’t” condition on being woken up. But if she does, she’ll get that strange result. If she does have cancer, she is more likely to be woken up multiple times than once, and being woken up at all does have some evidential weight.
All this, though, being merely verbal aids as I try to wrap my head around the consequences of the math. And therefore to be taken more circumspectly than the math itself.
If she does condition on being woken up, I think she still gets 1⁄2. I hate to keep repeating arguments, but what she knows when she is woken up is that she has been woken up at least once. If you just apply Bayes rule, you get 1⁄2.
If conditioning causes her to change her probability, it should do so in such a way that makes her more accurate. But as we see in the cancer problem, people with cancer give the same answer as people without.
Yes, but then we wouldn’t be talking about her credence on an awakening. We’d be talking about her credence on first waking and second waking. We’d treat them separately. With amnesia, 2 wakings are the same as 1. It’s really just one experience.
Apply it to what terms?
I’m not sure what more I can say without starting to repeat myself, too. All I can say at this point, having formalized my reasoning as both a Python program and an analytical table giving out the full joint distribution, is “Where did I make a mistake?”
Where’s the bug in the Python code? How do I change my joint distribution?
I like the version of your halfer variant version of your table. I still need to think about your distributions more though. I’m not sure it makes sense to have a variable ‘woken that day’ for this problem.