I raised this objection on Ksvanhorn’s initial post, though it came rather late, so I’m not sure if anyone saw it. You’ll have to forgive me in advance, as most of this math is beyond my current level of familiarity.
In the original post, Ksvanhorn states:
The statement of the problem implies that the distributions for PM and PT, conditional on H being false, are identical, but not necessarily independent.
My understanding is that Neal’s solution assumes that the sets of possible experience streams for Sleeping Beauty before answering the question are identical on both Monday and Tuesday. Furthermore, this “stream of experiences” includes events of arbitrarily small significance (one example given was the movement of a fly on the wall).
If my understanding is correct (and given that I don’t understand most of the math involved, it’s certainly possible that it’s not), it seems to be trivial to disprove this assumption. Through the course of the experiment, time passes. Sleeping Beauty ages. If something as insignificant as a fly on the wall or a change in heart rate is relevant enough to be included in these calculations, then Sleeping Beauty’s aging over the course of two should also be.
She cannot be two days older on Monday, and she cannot be one day older on Tuesday. All of her internal bodily functions proceed as normal. Her fingernails grow. Her hair grows. Any kind of wound condition that required healing will have progressed. Does she eat? Go to the bathroom? If we can condition on a nebulous “everything” that can include basically insignificant differences in experience, I can think of any number of much less insignificant differences that are affected by the passage of time, and thus cannot be identical on both Monday and Tuesday.
No, I very definitely do NOT assume that Beauty’s experiences are identical on Monday and Tuesday. I think one should solve the Sleeping Beauty problem with the ONLY fantastical aspect being the memory erasure. In every other respect, Beauty is a normal human being. If you then want to make various fantastic assumptions, go ahead, but thinking about those fantastic versions of the problem without having settled what the answer is in the usual version is unwise.
Just to clarify… ageing by one day may well be one reason Beauty’s experiences are different on Tuesday than on Monday, but we assume that other variation swamps ageing effects, so that Beauty will not be able to tell that it is Tuesday on this basis.
I understand that you do not assume Beauty’s experiences are identical on Monday and Tuesday. Rather, my understanding is that you assume that “the set of things it is possible for Beauty to experience on Monday” is identical to “the set of things it is possible for Beauty to experience on Tuesday”. Is my understanding incorrect?
Ah! I see I misread what you wrote. As you point out, it is implausible in real life that the set of possible experiences on Monday is exactly the same as the set of possible experiences on Tuesday, or at least it’s implausible that the probability distributions over possible experiences on Monday and on Tuesday are exactly the same. I think it would be fine to assume for a thought experiment that they are the same, however. The reason it would be fine is that you could also not assume they are the same, but just that they are very similar, which is indeed plausible, and the result would be that at most Beauty will obtain some small amount of information about whether it is Monday or Tuesday from what her experiences are, which will change her probability of the coin having landed Heads by only a small amount. Similarly, we don’t have to assume PERFECT memory erasure. And we don’t have to assume (as we usually do) that Beauty has exactly ZERO probability of dying after Monday and before she might have been woken on Tuesday. Etc, etc.
I raised this objection on Ksvanhorn’s initial post, though it came rather late, so I’m not sure if anyone saw it. You’ll have to forgive me in advance, as most of this math is beyond my current level of familiarity.
In the original post, Ksvanhorn states:
My understanding is that Neal’s solution assumes that the sets of possible experience streams for Sleeping Beauty before answering the question are identical on both Monday and Tuesday. Furthermore, this “stream of experiences” includes events of arbitrarily small significance (one example given was the movement of a fly on the wall).
If my understanding is correct (and given that I don’t understand most of the math involved, it’s certainly possible that it’s not), it seems to be trivial to disprove this assumption. Through the course of the experiment, time passes. Sleeping Beauty ages. If something as insignificant as a fly on the wall or a change in heart rate is relevant enough to be included in these calculations, then Sleeping Beauty’s aging over the course of two should also be.
She cannot be two days older on Monday, and she cannot be one day older on Tuesday. All of her internal bodily functions proceed as normal. Her fingernails grow. Her hair grows. Any kind of wound condition that required healing will have progressed. Does she eat? Go to the bathroom? If we can condition on a nebulous “everything” that can include basically insignificant differences in experience, I can think of any number of much less insignificant differences that are affected by the passage of time, and thus cannot be identical on both Monday and Tuesday.
No, I very definitely do NOT assume that Beauty’s experiences are identical on Monday and Tuesday. I think one should solve the Sleeping Beauty problem with the ONLY fantastical aspect being the memory erasure. In every other respect, Beauty is a normal human being. If you then want to make various fantastic assumptions, go ahead, but thinking about those fantastic versions of the problem without having settled what the answer is in the usual version is unwise.
Just to clarify… ageing by one day may well be one reason Beauty’s experiences are different on Tuesday than on Monday, but we assume that other variation swamps ageing effects, so that Beauty will not be able to tell that it is Tuesday on this basis.
I understand that you do not assume Beauty’s experiences are identical on Monday and Tuesday. Rather, my understanding is that you assume that “the set of things it is possible for Beauty to experience on Monday” is identical to “the set of things it is possible for Beauty to experience on Tuesday”. Is my understanding incorrect?
Ah! I see I misread what you wrote. As you point out, it is implausible in real life that the set of possible experiences on Monday is exactly the same as the set of possible experiences on Tuesday, or at least it’s implausible that the probability distributions over possible experiences on Monday and on Tuesday are exactly the same. I think it would be fine to assume for a thought experiment that they are the same, however. The reason it would be fine is that you could also not assume they are the same, but just that they are very similar, which is indeed plausible, and the result would be that at most Beauty will obtain some small amount of information about whether it is Monday or Tuesday from what her experiences are, which will change her probability of the coin having landed Heads by only a small amount. Similarly, we don’t have to assume PERFECT memory erasure. And we don’t have to assume (as we usually do) that Beauty has exactly ZERO probability of dying after Monday and before she might have been woken on Tuesday. Etc, etc.