Please show your work on your thirder Sleeping Beauty answer, given an observed random bit (α or β upon waking) that’s independent of the coin toss. I realize the bit is flipped on Tuesday, which is only observed by Beauty if tails, but a conditionally flipped independent random bit is still independent.
Why is the availability of a random bit necessary and/or sufficient to end the Beauty controversy?
(Sorry for the delay—I’m not notified when someone leaves a top-level comment.)
The halfer can try to argue like this: “SB doesn’t gain any information at all upon waking, because she was going to be woken whether the coin was heads or tails. Therefore, since the prior probabilities of heads and tails are 1⁄2 each, the posterior probabilities must also be 1⁄2 each.” I believe this was neq1′s argument in his “Sleeping Beauty Quips” post.
Then the thirder wants to say that the fact that SB gets woken twice in one branch but only once in the other means that, in some weird way, the probability of getting woken given tails was actually 2, rather than 1. So the likelihood function is Heads->1, Tails->2, and when we renormalise we get Heads->1/3 and Tails->2/3.
And naturally the halfer is skeptical, asking “How can you have a probability greater than 1???”
Hence, my devices of the uniform random number in [0,1] and the random bit are designed to ‘fix’ the thirder’s argument—in the sense of no longer having to do something that the halfer will think is invalid.
So now let’s consider that random bit.
We have four equally likely branches:
alpha = Monday, coin = Heads
alpha = Monday, coin = Tails
alpha = Tuesday, coin = Heads
alpha = Tuesday, coin = Tails
Now suppose SB wakes and sees ‘alpha’. Then SB now has the information “alpha is one the days on which I was woken”. This event is true in branches 1, 2 and 4 but false in branch 3.
In one of those three branches, the coin is heads, and in the other two it is tails. Hence we get our {1/3, 2⁄3} posterior probabilities for {heads, tails}.
You’re asking why this is necessary and/or sufficient to ‘end the controversy’. Necessary? Well, if we’re prepared to accept the original form of the thirder’s argument then of course this device isn’t necessary. Sufficient? Yes, unless there’s some reason why adding a random bit should alter what SB thinks the probabilities are.
Perhaps neq1 would place great importance on whether or not the Monday and Tuesday instances of SB had exactly the same or slightly different mental states, and say that without the random bit, what we have when SB is woken twice is really a single conscious observer whose mind is split over two days. Personally, I think this is absurd, because sameness or slight-difference of minds has no effects on anything we can observe and is therefore impossible to verify or falsify. (Consider that even if we try to keep the room exactly the same on Tuesday, the Moon will be in a slightly different position, and so SB will be affected differently by its gravity. Perhaps that puts her into a “slightly different” mental state?)
Please show your work on your thirder Sleeping Beauty answer, given an observed random bit (α or β upon waking) that’s independent of the coin toss. I realize the bit is flipped on Tuesday, which is only observed by Beauty if tails, but a conditionally flipped independent random bit is still independent.
Why is the availability of a random bit necessary and/or sufficient to end the Beauty controversy?
(Sorry for the delay—I’m not notified when someone leaves a top-level comment.)
The halfer can try to argue like this: “SB doesn’t gain any information at all upon waking, because she was going to be woken whether the coin was heads or tails. Therefore, since the prior probabilities of heads and tails are 1⁄2 each, the posterior probabilities must also be 1⁄2 each.” I believe this was neq1′s argument in his “Sleeping Beauty Quips” post.
Then the thirder wants to say that the fact that SB gets woken twice in one branch but only once in the other means that, in some weird way, the probability of getting woken given tails was actually 2, rather than 1. So the likelihood function is Heads->1, Tails->2, and when we renormalise we get Heads->1/3 and Tails->2/3.
And naturally the halfer is skeptical, asking “How can you have a probability greater than 1???”
Hence, my devices of the uniform random number in [0,1] and the random bit are designed to ‘fix’ the thirder’s argument—in the sense of no longer having to do something that the halfer will think is invalid.
So now let’s consider that random bit.
We have four equally likely branches:
alpha = Monday, coin = Heads
alpha = Monday, coin = Tails
alpha = Tuesday, coin = Heads
alpha = Tuesday, coin = Tails
Now suppose SB wakes and sees ‘alpha’. Then SB now has the information “alpha is one the days on which I was woken”. This event is true in branches 1, 2 and 4 but false in branch 3.
In one of those three branches, the coin is heads, and in the other two it is tails. Hence we get our {1/3, 2⁄3} posterior probabilities for {heads, tails}.
You’re asking why this is necessary and/or sufficient to ‘end the controversy’. Necessary? Well, if we’re prepared to accept the original form of the thirder’s argument then of course this device isn’t necessary. Sufficient? Yes, unless there’s some reason why adding a random bit should alter what SB thinks the probabilities are.
Perhaps neq1 would place great importance on whether or not the Monday and Tuesday instances of SB had exactly the same or slightly different mental states, and say that without the random bit, what we have when SB is woken twice is really a single conscious observer whose mind is split over two days. Personally, I think this is absurd, because sameness or slight-difference of minds has no effects on anything we can observe and is therefore impossible to verify or falsify. (Consider that even if we try to keep the room exactly the same on Tuesday, the Moon will be in a slightly different position, and so SB will be affected differently by its gravity. Perhaps that puts her into a “slightly different” mental state?)