The Beauty doesn’t know only about one pass she knows about their relation as well. And because of it she can’t reason as if they happen at random. You need to address this point before we could move on, because all your further reasoning is based on the incorrect premise that beauty knows less than she actually knows.
She has no ability to infer/anticipate what the coins were/will be showing on another day.
She absolutely has this ability as long as she knows the procedure, that TT and TH follow in pairs, she can make such conditional statements: “if the coins are currently TT then they either will be TH tomorrow or were TH yesterday”. It’s very different from not knowing anything whatsoever about the state of the coin on the next day. The fact that you for some reason feel that it should not matter is irrelevant. It’s still clearly more than no information whatsoever and, therefore, she can’t justifiably reason as if she doesn’t have any.
On the other hand, if the memory wipe removed this knowledge from her head as well, if the only thing she truly knew was that she is currently awakened at one of three possible states either TH, HT and TT, and had no idea of the relationship between them, then, only then, she would be justified to reason as you claim she should.
What you are doing, is treating HH (or, in Elga’s implementation, H&Tuesday) as if it ceases to exist
No, I treat is as an event that Beauty doesn’t expect to observe and therefore she doesn’t update when she indeed doesn’t observe it according to the law of conservation of expected evidence. We are talking about Beauty’s perspective after all, not a some outside view.
Suppose an absolutely trustwothy source tells you that the coin is Heads side up. Then you go and look at the coin and indeed it’s Heads side up. What should have been your probability that the coin is Tails side up before you looked at it?
It should be zero. You’ve already known the state of the coin before you looked at it, you got no new information. Does it mean that Tails side of a coin doesn’t exist? No, of course not! It just that you didn’t expect that the coin could possibly be Tails in this particular case based on your knowledge state.
Say I roll a six-sided die tell you that the result is odd. Then I administer the amnesia drug, and tell you that I previously told you whether th result was even or odd. I then ask you for your degree of belief that the result is a six. Should you say 1⁄6, because as far as you know the sample space is {1,2,3,4,5,6}? Or should you say 0, because “you are [now] observing a state that you’ve already observed is only {1,3,5}?
I was going to post a generalized way of reasoning under amnesia in a future post, but here is some: getting memory erased about some evidence just brings you to the state where you didn’t have this particular evidence. And getting an expected memory wipe can only make you less confident in your probability estimate, not more.
In this dice rolling case, initially my P(6) = 1⁄6, then after you told me that it’s odd, P(6|Odd)=0, and then when I’m memory wiped I’m back to P(6) = 1⁄6 and the knowledge that you’ve already told me whether the result is even or odd doesn’t help P(6|Even or Odd) = 1⁄6
Likewise in Sleeping Beauty I initially have P(Heads) = 1⁄2. Then I awakened exactly as I’ve expected in the experiment and still have P(Heads|Awake) = 1⁄2. Now suppose that I’m awakened once more. If there was no memory wipe I’d learn that I’m a awake a second time which would bring me to P(Heads|Two Awakenings) = 0. But I do not get this evidence due to memory wipe. So due to it, when I’m awakened the second time, I once again learn that I’m awake and still having P(Heads|Awake) = 1⁄2.
What you are implicitly claiming, however, is that getting memory wiped, or even just a possibility of it, makes the Beauty more confident in one outcome over the other! Which is quite bizarre. As if knowing less gives you more knowledge. Moreover, you assume that the person who knowns that their memory was/may be erased, just have to act as if they do not know it.
Suppose a coin is tossed and you received some circumstantial evidence about it’s state. As a result you are currently at 2⁄3 in favor of Heads. And then I tell you: “What odds are you ready to bet on? By the way, I have erased from your memory some crucial evidence in favor of Tails”. Do you really think that you are supposed to agree to bet on 1:2 odds even though you now know that the state of the evidence your currently have may not be trustworthy?
This is the crux of our disagreement.
The Beauty doesn’t know only about one pass she knows about their relation as well. And because of it she can’t reason as if they happen at random. You need to address this point before we could move on, because all your further reasoning is based on the incorrect premise that beauty knows less than she actually knows.
She absolutely has this ability as long as she knows the procedure, that TT and TH follow in pairs, she can make such conditional statements: “if the coins are currently TT then they either will be TH tomorrow or were TH yesterday”. It’s very different from not knowing anything whatsoever about the state of the coin on the next day. The fact that you for some reason feel that it should not matter is irrelevant. It’s still clearly more than no information whatsoever and, therefore, she can’t justifiably reason as if she doesn’t have any.
On the other hand, if the memory wipe removed this knowledge from her head as well, if the only thing she truly knew was that she is currently awakened at one of three possible states either TH, HT and TT, and had no idea of the relationship between them, then, only then, she would be justified to reason as you claim she should.
No, I treat is as an event that Beauty doesn’t expect to observe and therefore she doesn’t update when she indeed doesn’t observe it according to the law of conservation of expected evidence. We are talking about Beauty’s perspective after all, not a some outside view.
Suppose an absolutely trustwothy source tells you that the coin is Heads side up. Then you go and look at the coin and indeed it’s Heads side up. What should have been your probability that the coin is Tails side up before you looked at it?
It should be zero. You’ve already known the state of the coin before you looked at it, you got no new information. Does it mean that Tails side of a coin doesn’t exist? No, of course not! It just that you didn’t expect that the coin could possibly be Tails in this particular case based on your knowledge state.
I was going to post a generalized way of reasoning under amnesia in a future post, but here is some: getting memory erased about some evidence just brings you to the state where you didn’t have this particular evidence. And getting an expected memory wipe can only make you less confident in your probability estimate, not more.
In this dice rolling case, initially my P(6) = 1⁄6, then after you told me that it’s odd, P(6|Odd)=0, and then when I’m memory wiped I’m back to P(6) = 1⁄6 and the knowledge that you’ve already told me whether the result is even or odd doesn’t help P(6|Even or Odd) = 1⁄6
Likewise in Sleeping Beauty I initially have P(Heads) = 1⁄2. Then I awakened exactly as I’ve expected in the experiment and still have P(Heads|Awake) = 1⁄2. Now suppose that I’m awakened once more. If there was no memory wipe I’d learn that I’m a awake a second time which would bring me to P(Heads|Two Awakenings) = 0. But I do not get this evidence due to memory wipe. So due to it, when I’m awakened the second time, I once again learn that I’m awake and still having P(Heads|Awake) = 1⁄2.
What you are implicitly claiming, however, is that getting memory wiped, or even just a possibility of it, makes the Beauty more confident in one outcome over the other! Which is quite bizarre. As if knowing less gives you more knowledge. Moreover, you assume that the person who knowns that their memory was/may be erased, just have to act as if they do not know it.
Suppose a coin is tossed and you received some circumstantial evidence about it’s state. As a result you are currently at 2⁄3 in favor of Heads. And then I tell you: “What odds are you ready to bet on? By the way, I have erased from your memory some crucial evidence in favor of Tails”. Do you really think that you are supposed to agree to bet on 1:2 odds even though you now know that the state of the evidence your currently have may not be trustworthy?