The question is about what information you actually have.
In the linked example, it may seem that you have precisely the information “there is at least one heads.” But if you condition on that you get the wrong answer. The explanation is that, in this type of memory loss situation, waking up and experiencing y is not equivalent to “I experience y at least once.” When you wake up and experience y you do know that you must experience y on either monday or tuesday, but your information is not equivalent to that statement.
If you asked on sunday “will I experience y at least once?” then the answer would be relevant. But if we nailed down the precise information gained from waking up and experiencing y, it would be irrelevant.
Beauty’s information isn’t “there is at least one head”, but rather, “there is a head seen by Beauty on a day when her nose itches, the fly on the wall is crawling upwards, her sock is scrunched up uncomfortably in her left shoe, the sharp end of a feather is sticking out of one of the pillows, a funny tune she played in high school band is running though her head, and so on, and so on.”.
I’m talking about the method you’re using. It looks like when you wake up and experience y you are treating that as equivalent to “I experience y at least once.”
This method is generally incorrect, as shown in the example. Waking up and experiencing y is not necessarily equivalent to “I experience y at least once.”
If you yourself believe the method is incorrect when y is “flip heads”, why should we believe it is correct when y is something else?
After my other response to this, I thought a bit more about the scenario described by Conitzer. A completely non-fantastic version of this would be as follows (somewhat analogous to my Sailor’s Child problem, though the whole child bit is not really necessary here):
You have two children. At age 10, you tell both of them that their Uncle has flipped two coins, one associated with each child, though the children are told nothing that would let them tell which is “their” coin. When they turn 20, they will each be told how “their” coin landed, in two separate rooms so they will not be able to communicate with each other.. They will then be asked what the probability is that the two coin flips were the same. (The two children correspond to two awakenings of Beauty.)
If you are one of these children, and are told that “your” coin landed heads, what should you give for the probability that the two flips are the same? It’s obvious that the correct answer is 1⁄2. But you might argue that their are four equally-likely possibilities for the two flips—HH, HT, TH, and TT—and that observing a head eliminates TT, giving a 1⁄3 probability that the two flips are the same.
This is of course an elementary mistake in probabilistic reasoning, caused by not using the right space of outcomes. Suppose that one of the children is left-handed and one is right-handed. Then there are actually eight equally-likely possibilities—RHH, LHH, RHT, LHT, RTH, LTH, RTT, LTT—where the initial R or L indicates whether the first coin is for the right-handed child or the left-handed child. Suppose you are the right-handed child. Observing heads eliminates LHT, RTH, RTT, and LTT, with the remaining possibilities being RHH, LHH, RHT, and LTH, in half of which the flips are the same. So the answer is now seen to be 1⁄2.
But why is this the right answer to this non-fantastical problem? (I take it that it is correct, and that this is not controversial.) The reason is that we know how probability works in ordinary situations, in which personal identities are clear, because everyone has different experiences. If instead we make a fantastic assumption that the two children are identical twins raised apart in absolutely identical environments, and therefore have exactly the same thoughts and experiences, up until the point at age 20 when they are told possibly-different things about how their coins landed, it may not be so clear that 1⁄2 is the right answer. It’s also not so clear that this fantastic scenario is possible, or of interest. It certainly would not be a good idea to treat it as being just the same as the non-fantastical scenario, apart from a little simplifying assumption about identical experiences...
In any not-completely-fantastical scenario, Beauty’s experiences on Monday are very unlikely to be repeated exactly on Tuesday, so “experiences y” and “experiences y at least once” are effectively equivalent. Any argument that relies on her sensory input being so restricted that there is a substantial probability of identical experiences on Monday and Tuesday applies only to a fantastical version of the problem. Maybe that’s an interesting version of the problem (though maybe instead it’s simply an impossible version), but it’s not the same as the usual, only-mildly-fantastical version.
The question is about what information you actually have.
In the linked example, it may seem that you have precisely the information “there is at least one heads.” But if you condition on that you get the wrong answer. The explanation is that, in this type of memory loss situation, waking up and experiencing y is not equivalent to “I experience y at least once.” When you wake up and experience y you do know that you must experience y on either monday or tuesday, but your information is not equivalent to that statement.
If you asked on sunday “will I experience y at least once?” then the answer would be relevant. But if we nailed down the precise information gained from waking up and experiencing y, it would be irrelevant.
Beauty’s information isn’t “there is at least one head”, but rather, “there is a head seen by Beauty on a day when her nose itches, the fly on the wall is crawling upwards, her sock is scrunched up uncomfortably in her left shoe, the sharp end of a feather is sticking out of one of the pillows, a funny tune she played in high school band is running though her head, and so on, and so on.”.
I’m talking about the method you’re using. It looks like when you wake up and experience y you are treating that as equivalent to “I experience y at least once.”
This method is generally incorrect, as shown in the example. Waking up and experiencing y is not necessarily equivalent to “I experience y at least once.”
If you yourself believe the method is incorrect when y is “flip heads”, why should we believe it is correct when y is something else?
After my other response to this, I thought a bit more about the scenario described by Conitzer. A completely non-fantastic version of this would be as follows (somewhat analogous to my Sailor’s Child problem, though the whole child bit is not really necessary here):
You have two children. At age 10, you tell both of them that their Uncle has flipped two coins, one associated with each child, though the children are told nothing that would let them tell which is “their” coin. When they turn 20, they will each be told how “their” coin landed, in two separate rooms so they will not be able to communicate with each other.. They will then be asked what the probability is that the two coin flips were the same. (The two children correspond to two awakenings of Beauty.)
If you are one of these children, and are told that “your” coin landed heads, what should you give for the probability that the two flips are the same? It’s obvious that the correct answer is 1⁄2. But you might argue that their are four equally-likely possibilities for the two flips—HH, HT, TH, and TT—and that observing a head eliminates TT, giving a 1⁄3 probability that the two flips are the same.
This is of course an elementary mistake in probabilistic reasoning, caused by not using the right space of outcomes. Suppose that one of the children is left-handed and one is right-handed. Then there are actually eight equally-likely possibilities—RHH, LHH, RHT, LHT, RTH, LTH, RTT, LTT—where the initial R or L indicates whether the first coin is for the right-handed child or the left-handed child. Suppose you are the right-handed child. Observing heads eliminates LHT, RTH, RTT, and LTT, with the remaining possibilities being RHH, LHH, RHT, and LTH, in half of which the flips are the same. So the answer is now seen to be 1⁄2.
But why is this the right answer to this non-fantastical problem? (I take it that it is correct, and that this is not controversial.) The reason is that we know how probability works in ordinary situations, in which personal identities are clear, because everyone has different experiences. If instead we make a fantastic assumption that the two children are identical twins raised apart in absolutely identical environments, and therefore have exactly the same thoughts and experiences, up until the point at age 20 when they are told possibly-different things about how their coins landed, it may not be so clear that 1⁄2 is the right answer. It’s also not so clear that this fantastic scenario is possible, or of interest. It certainly would not be a good idea to treat it as being just the same as the non-fantastical scenario, apart from a little simplifying assumption about identical experiences...
In any not-completely-fantastical scenario, Beauty’s experiences on Monday are very unlikely to be repeated exactly on Tuesday, so “experiences y” and “experiences y at least once” are effectively equivalent. Any argument that relies on her sensory input being so restricted that there is a substantial probability of identical experiences on Monday and Tuesday applies only to a fantastical version of the problem. Maybe that’s an interesting version of the problem (though maybe instead it’s simply an impossible version), but it’s not the same as the usual, only-mildly-fantastical version.