Sleeping Beauty problem is probably the worst possible example for this kind of observation. Consider a much less controversial problem, isomorphic to Single Awakening problem from the post:
A coin is tossed. On Heads you are given a red ball. On Tails you are given either red or blue ball. You got a red ball. What should be your credence that the coin is Heads?
There are two different probabilities:
Unconditional probability that the coin is Heads: P(Heads) = 1⁄2
Probability that the coin is Heads conditional on getting a red ball: P(Heads|Red) = 2⁄3
Both of these probabilities have frequentist meaning. If you run the experiment some large number of times in about 1⁄2 of all iterations the coin will be Heads, but if you count only the iterations where you receive a red ball, the coin is Heads in about 2⁄3 of such iterations. This is possible because number of iterations where you receive a red ball is lower than the total number of iterations.
Your credence in the event is probability of this events conditional on all the events you know has happened in this iteration of the experiment. So before you receive the red ball your credence is 1⁄2, and after receiving it, it’s 2⁄3.
A lot of people mistakenly assume that the same thing is going with Sleeping Beauty problem, but it’s not the case. There are no such iterations of the experiment where the Beauty is not awake in the experiment so both P(Heads) and P(Heads|Awake) have to be equal. So both before the awakening in the experiment and after your credence stays the same—its 1⁄2. I talk more about it in the next posts in the series.
For example, you toss the coin and speak with one random person if Heads, but with two if Tails. In that case I—if approached by you—can expect that there is 2⁄3 that you have Tail coin even before you tell me how it fails.
Yes, but only if I know that I’m a random person. If there are two people who you can talk to, and on Heads you talk to a random one and on Heads you talk to both, then in 2⁄3 iterations of the experiment when you talk to me its Tails, so I update my credence.
But suppose that on Heads you always talk to me and on Tails you talk both to me and to the other person. Then the fact that you approached me tells me nothing about the state of the coin and I should not change my credence.
Thanks for your detailed answer. I think that for halfer the most interesting part is than the SB learns that it is Monday. This provides her with some information about the current toss results, namely that it is 1⁄4 for Tails. Do you address this problem in the next posts?
Sleeping Beauty problem is probably the worst possible example for this kind of observation. Consider a much less controversial problem, isomorphic to Single Awakening problem from the post:
There are two different probabilities:
Unconditional probability that the coin is Heads: P(Heads) = 1⁄2
Probability that the coin is Heads conditional on getting a red ball: P(Heads|Red) = 2⁄3
Both of these probabilities have frequentist meaning. If you run the experiment some large number of times in about 1⁄2 of all iterations the coin will be Heads, but if you count only the iterations where you receive a red ball, the coin is Heads in about 2⁄3 of such iterations. This is possible because number of iterations where you receive a red ball is lower than the total number of iterations.
Your credence in the event is probability of this events conditional on all the events you know has happened in this iteration of the experiment. So before you receive the red ball your credence is 1⁄2, and after receiving it, it’s 2⁄3.
A lot of people mistakenly assume that the same thing is going with Sleeping Beauty problem, but it’s not the case. There are no such iterations of the experiment where the Beauty is not awake in the experiment so both P(Heads) and P(Heads|Awake) have to be equal. So both before the awakening in the experiment and after your credence stays the same—its 1⁄2. I talk more about it in the next posts in the series.
Yes, but only if I know that I’m a random person. If there are two people who you can talk to, and on Heads you talk to a random one and on Heads you talk to both, then in 2⁄3 iterations of the experiment when you talk to me its Tails, so I update my credence.
But suppose that on Heads you always talk to me and on Tails you talk both to me and to the other person. Then the fact that you approached me tells me nothing about the state of the coin and I should not change my credence.
Thanks for your detailed answer. I think that for halfer the most interesting part is than the SB learns that it is Monday. This provides her with some information about the current toss results, namely that it is 1⁄4 for Tails. Do you address this problem in the next posts?
In the next part I drive the correct Halfer model that neither updates on awakening nor on learning that it’s Monday.