I think that by the time you edit the example enough so that it works the way they want it to, you will have something so contrived that it is nearly as absurd as a story about a fairy tale princess in an amnesia experiment. And then it would be completely unremarkable to see an absurd example producing an absurd conclusion.

I would like to see them present this example completely rigorously, explicitly identifying J, as well as all of the objective propositions in the probability space. I am confident enough in the reasonableness of formal logic and probability to say that if they did this, then we would see that either their reasoning is flawed, or their premises are ridiculous. (This same theme is evident in the distinction between J'' and J* in my discussion with PairTheBoard.)

But in the SB situation, we have more than just a prior of .5, we are actually assuming that the coin is fair and so there is an objective probability of 50% that the coin is going to be heads. In the T1/T2 situation we have a prior of .5, but we don't actually know that both are genuinely possible.

EDIT: In terms of your credence for believing in one or the other I see your point, since all that matters in that situation would be the priors, I guess. I don't really have any training here, but I feel like there's a genuine difference between the two situations.

Seems like semantics to me. I'll go with a version of a suggestion someone else made:

What are the chances that a tossed coin lands heads?=1/2
What are they chances that you, Sleeping Beauty,are being interviewed when the coin was heads up?=1/3.

Can someone explain this for me? The math he uses with the centered assumption "Today is Monday":

Looks like this is intended to be an alternate rebuttal of the 1/2 logic that excludes the "unfair bet" issue. If someone can clarify on "13" it would be great.

Please don't bother if you're just going to say he's wrong or made a mistake without a thorough run through

http://www.anthropic-principle.com/p.../synthesis.pdf

The original 1/2 view logic, which Bostrom's argument attacks:

'Beautys credence for the coinflip on any particular random awakening in the experiment is always 50/50 heads/tails. But if SB is informed of the fact that it is Monday, then her credence of heads should be updated to 2/3 (for various objectionable reasons detailed in the link I posted).'

It seems then that if 1/2ers don't accept updating SB's credence of heads based on being told "Today is Monday", then the point is moot. So apparently Bostrom is just illustrating why this logic fails. That being said, "Today is Tuesday" would irrefutably inform Beauty that she needs to update her credence of tails to 100%.*

Uncertain of whether the 1/2 viewpoint demands updated credence once one is told it is Monday, as Lewis states. I highly doubt it and it seems that Bostrom is probably just picking apart David Lewis' logic here. This is an important point though at the root of this question, imo.

*It seems potentially relevant to me that this new information SB has gained regarding the result of the coin flip, would also have been implied in telling SB "You have been awoken before". This statement also informs SB to how her previous bet turned out, while just telling her "today is Monday" doesn't imply multiple awakenings or the result of the coin flip. So maybe these are two very different statements in terms of self-locating information that could update credence. Or not.

This was a fun thread. Jason1990 mentioned it recently; Veritasium did a video on it recently. I didn't find his analysis particularly insightful.

Flip a fair coin. I believe the probability of it coming up heads is 1/2. But if betting a dollar on heads commits me to making the bet twice when I'm wrong then I want 2-1 odds on the bet(s).


PairTheBoard

All that means is that you lose 2 units when you're wrong and gain 1 unit when you're right. In other words, you're getting 1:2 odds, which is insufficient obviously. You don't want 2:1. You want 1:1 and you're not getting it.

So, having a preference for betting on heads, when SB is asked to bet on awakening rather than changing her 50-50 belief for the coin flip she should realize she's not getting 1-1 odds on her bet(s).


PairTheBoard

1/3.

Man, 12 years just whizzed by.

I enjoyed the Veritasium video and all the thread posts. My feeling is that either answer can be correct. I say that because the way the problem is presented is misleading. Both the question being asked of Sleeping Beauty and the Sleeping Beauty's answer are ambiguously stated. The question "What is your credence now for the proposition that the coin landed heads?" is asked of her at each interview, twice when tails occurs. It is unclear whether the question is meant to assess SB’s belief in the true rate of Heads occurrences or, instead, to assess the expected rate of her correct guesses of Heads. Assume that Sleeping Beauty is mathematically astute. At the interviews she could answer “generally speaking the chances of Heads will always be ½ “ and she would be correct. Or she could say “Since you are counting all my guesses, the chances of my guess of “Heads” being correct for this specific coin toss is 1/3 and she would also be correct.

I hope I can post again in another 12 years!

There are three possible awakening situations. HM=Heads on Monday, TM=Tails on Monday, and TT=Tails on Tuesday. I think the problem lies in how an awakened SB answers the question, "What is my credence for which awakening I'm in?"

(1/3) She could reason, "My possible awakenings are indistinguishable so I might as well consider them equally likely. But only one is due to Heads so the chance I'm in an awakening due to Heads is 1/3."

(1/2) Or she could reason, "Half the time the coin is heads so there's half a chance I'm in the one and only awakening due to Heads. Half the time the coin is tails so there's half a chance I'm in one or the other of the Tails awakenings. If I am in one of the Tails awakenings I might as well consider Monday and Tuesday equally likely. So HM=1/2, TM=1/4, TT=1/4".

A problem with (1/2) is if you tell SB, "This is Monday". Conditional probability seems to force her to conclude her awakening is twice as likely due to Heads than to Tails. Yet that's absurd since the coin could just as well wait until Tuesday morning to be flipped.

A problem with (1/3) is that the assumption of "equally likely" seems arbitrary. How could you test it? How could you make use of the Game as played to produce the outcome of a "random awakening" where the frequencies would be 1/3 each. You could spin a 3-way spinner. But it would be inconvenient if the spinner said, "look at the Heads on Monday awakening" when in the actual game the coin comes up Tails.

The Game could be used to produce a random awakening that agrees with the HM=1/2, TM=1/4, TT=1/4. Just flip a Quarter for Heads or Tails and if Tails flip a Dime for Monday or Tuesday. Trouble is, flipping the Dime is not really part of the Game. It's an arbitrary add-on.

Consider this. Ask someone to pick a random integer. What's your credence for her picking, say, 716? You can't resort to an "equally likely" refuge since an "equally likely" probability distribution for the integers is impossible. Yet she does pick a number "at random" and writes it down. What's your credence for it? Maybe your best answer is, "Can't say".

Maybe that's what SB should say. Or maybe she can say it's equally likely she's in the process produced by Heads and the process produced by Tails. That remains true if she's told it's Monday. If told it's Heads she knows it's Monday. If told it's Tuesday she knows it's Tails. And if told it's Tails she says she can't say what part of the Tails process she's in.


PairTheBoard

PairTheBoard

If she doesn't like money she could.

Nobody likes laying 2-1 on a 50-50 proposition.


PairTheBoard

I just read this entire thread. Reading all those posts by PairTheBoard was like reading posts made by someone else. Maybe the GOAT SMP thread. Can't say the above post by cyberfish was the best in the thread but in some ways it's my favorite. I don't think it got any responses. In his scenario for the After Experiment...

I see Sleeping Beauty awake in the experiment. She has a 20 sided die in her hand. She is noticing every detail of what's happening where she's at. She's just read all the professional philosopher 1/3er papers. She's reading this thread and studying all the great arguments for why she should reduce her 1/2 heads credence to 1/3. And she thinks to herself,

"After listening to all these really smart people, I know I should reduce my credence for heads having been flipped from 1/2 to 1/3. Trouble is, I know damn well that when this silly experiment is over there's still a 50-50 chance I'm going into the Fridge."


PairTheBoard