Well, modulo temporal deixis, they are the same thing… At any rate, when you wake up, you don’t know what day it is. So what’s the proposition that you’re saying you’ve learnt? Given that “the coin flip came up heads at the start of the experiment” is a proposition which lacks temporal deixis (i.e., the truth value of this statement doesn’t vary depending on when it’s made, as it makes no reference to “now” or “tomorrow” or any such relative terms), if one changes the probabilities assigned to this, it seems like one must have learnt at least some similarly non-deictic fact.
Blaster Master, the Monty Hall paradox does turn upon learning more information than one thinks. When Monty Hall says to you “There is no prize behind Door B”, you learn two things: that there is no prize behind Door B, and that Monty Hall chooses to tell you of Door B that there is no prize behind it. If you knew only the first, then the odds for switch or stay would be the same; it is the fact that you learn the second as well which makes switch more attractive.
True, and I don’t want to stretch the analogy too far, but when you think about it, you don’t really learn anything when he reveals a door because you knew he would reveal a door before the game started. You also know that, regardless of whether you chose correctly or not, that at least one door that you didn’t choose would not have the prize, and so, as long as he opens a door with no prize behind it, whether he opened door #2 or door #3, you’re still in the same situation.
Basically, my point was that you’re not actually learning anything because nothing happens that makes any of the potential outcomes distinguishable to the player. You’re not really learning anything because, it wouldn’t matter which door he openned, your optimal solution remains unchanged regardless.
It’s the same here with the Sleeping Beauty Paradox. You’re not really gaining any information that you didn’t have available before the experiment happened because SB doesn’t have any way to distinguish what day it is. The difference is, she knows that she’d be woken up at a different frequency based upon the outcome of the coin toss, which clearly means that the waking event is not independent of the coin toss, which means the dependent probability of the coin toss CANNOT be 1/2, or it would be an independent event, which is clearly a contradiction.
The comparison I was trying to make is that in the Monty Hall problem, is that the intuitive probability doesn’t make sense becasue you’re calculating a probabilty for a different circumstance. In Monty Hall, you’re not calculating for just the door you have and the door you don’t, but for the door you have and both doors you don’t. In this scenario, you’re not calculating the probability of a coin landing heads, you’re calculating the probabilty that it was heads given that some event that depends upon the outcome of the coin flip occurred.
Another way of looking at it is to change the interview, so that the interview consists of the question, “What is your credence now for the proposition that today is Tuesday?” And the correct answer for that is, one chance in three, because that’s one of three equally likely scenarios.
Not too interesting, from my point of view. An event has an equal likelihood of occurring, but the outcome is essentially doubled for one of the two outcomes.
Not a paradox, IMO.
If I take a picture of you every day in January, and twice a day in March, then randomly show you a picture I took, are you surprised the odds are 2/3 it’s from March? Nobody would be impressed that I say January and March have the same number of days, either.
Let’s assume the experiment is heads=>one day of sleep, tails=>ninety-nine days of sleep (instead of two). When Beauty is woken in this version, how confident can she be that the coin was heads?
Tyrrell, I’ve been thinking about your random calendar business, and I’m not sure how I feel about it. Upon awakening, prior to taking a look at her random calendar, what is it that you would say SB’s credence is for the coin being heads? 1/2 or 1/3? Clearly, after looking at the random calendar, no matter what, her credence will be, on your account, 1/3; but since this occurs no matter which symbol is on the calendar, if credences follow probabilistic rules of conditioning, then her credence prior to observing the calendar must also be 1/3.
But if the mere existence of a random calendar, whether or not SB has yet been able to look at it, causes her credence to be 1/3, then how can there be any context in which her credence is 1/2? Lots of calendars exist, random and otherwise; it’s just that, in some setups, SB, upon awakening, will be interrogated before she has the chance to look at any of them.
Or, to put it another way, both this problem and the Monte Hall problem have a situation where there are two probabilities that people tend to think should have to be equal but they are not because of the way you have gotten to that point. In this problem, the two probabilities in question are the probability that, when queried, Sleeping Beauty should conclude that the coin landed heads vs tails. In the Monte Hall problem, the two probabilities in question are the probabilities that the prize is behind Door #1 vs Door #3 once Monte Hall has shown you that it is not behind Door #2 (assuming he has followed the protocol discussed in an unambiguous statement of that problem).
And, that of course, is very analogous to making the Monte Hall problem involve 100 doors and imagining that after you pick your door, Monte Hall opens up 98 of the other 99 to reveal the lack of a prize. In that case, it becomes clearer to most people that they should switch to the one remaining door.
I had the same thought, but it leads me to the opposite conclusion. That is, why on earth would you think that a fair coin had a 1/100 chance of coming up heads? The experiment itself has no effect on the coin’s flipping, and she know that a tails will eventually lead to her awakening.
The paradox appears in that, if you play everything out, she’ll be wrong far more often by guessing tails than by guessing heads–but the fact of her guess has nothing to do with the fact of the coin’s flipping. It’s just that, on a tails, she’ll get a lot more opportunities to make guesses.
She would then effectively be in the calendar-less case, where I’m a halfer: She should have credence 1/2.
Not quite. Your premise, as I read it, is the following:
If SB should have credence of 1/3 in HEADS after looking at the calendar, no matter what it displays, then, prior to looking at the calendar, SB should give equal credence to each of the following:
[ul]
[li]HEADS given that the calendar is displaying symbol 1 today,[/li][li]HEADS given that the calendar is displaying symbol 2 today,[/li][li]HEADS given that the calendar is displaying symbol 3 today,[/li][li] . . . etc.[/li][/ul]
The way that I block this is that, on my view, SB has no way to make these strings of words into meaningful statements until she looks at the calendar (or something equivalent). Until then, they are not statements in which it makes sense to have a credence. Prior to looking at the calendar, the word “today” can’t refer to anything. It is only after she learns what the calendar displays (or something equivalent) that she can give “today” meaning as “the day on which the calendar displays <blah>”.
It’s not the mere existence of a random calendar that changes her credence. (I agree that equivalents of such calendars are almost certainly abundant). She needs in addition to know the specific symbol that the random calendar displays, as distinct from all the other symbols that it could display. Only then should her credence be computed by conditioning on the calendar displaying that particular symbol during the experiment. Otherwise, she should only condition on the calendar showing some symbol during the experiment, which gives a different result.
This distinction is what I was trying to highlight with my cupcake paradox from the earlier thread. There’s a difference between conditioning on
[ul]
[li] some symbol appears on the calendar during the experiment, vs[/li][li] that symbol appears on the calendar during the experiment.[/li][/ul]
As readers of the previous thread will recall, my cupcake paradox didn’t make quite the epistemic point that I’d hoped. But, without rehashing that thread, I just want to point out that the objection that arose there doesn’t apply to my approach to the SB paradox.
If one were to try to apply that objection, one would argue that SB should condition on neither of the above. Rather (one would argue) she should condition on
[ul]
[li] that symbol, instead of one of the other symbols that appears during the experiment, appears on the calender today.[/li][/ul]
However, my approach forbids this move because, on my view, she has no way to indicate today other than as the day on which that particular symbol appears.
Just to make it clear that my “credences follow probabilistic rules of conditioning”:
After looking at the random calender and seeing symbol <blah>, SB’s credence in HEADS should be the probability of the conditional statement
[ul]
[li]HEADS given that the calendar displays symbol <blah> during the experiment.[/li][/ul]
Furthermore, prior to looking at the random calendar, SB would indeed assign credence of 1/3 to each of the following conditionals:
[ul]
[li]HEADS given that the calendar displays symbol 1 during the experiment,[/li][li]HEADS given that the calendar displays symbol 2 during the experiment,[/li][li]HEADS given that the calendar displays symbol 3 during the experiment,[/li][li] . . . etc.[/li][/ul]
So there is no violation of the rules of probability here. However, this doesn’t make her a thirder prior to looking at the calendar. For each of the conditionals above, she does not yet know whether its condition obtains. Therefore, she shouldn’t use any of them to compute her credence in HEADS.
The coin has a 50-50 chance of coming up heads or tails. But, the question isn’t “what is the probability the coin was heads”, it’s “what is the probability the coin was heads, given that she was just woken up”. It’s a subtle difference, but it dramatically changes the calculation. And “on a tails, she’ll get a lot more opportunities to make guesses” is exactly why the answer is 1/100 (or 1/3 in the original).
If she answers heads, she will be wrong 99 out of 100 times she is woken up.
But wouldn’t you say that the following is a better analogy? I flip a coin. If I get heads, I take 1 picture per day of you in January (but none in March). If I get tails, I take 2 pictures per day of you in March (but none in January). If I randomly show you a picture I took, are you surprised the odds are 1/2 it’s from March?
Alright, good point. Sorry, it should have been clear that, on your account, SB could no more say “The symbol on the calendar under my thumb now is…” than she could “Today is Monday”.
So instead I return to the question, why is SB not able to have a belief in “The symbol on the calendar under my thumb is…”? And, specifically, why would such reasons not apply equally as well to her ability have a belief in “The value of the coin flip (kept under my other thumb) is…”?
Incidentally, I may as well give my view on the paradox: I don’t think there is some general, unqualified notion of “the correct probability distribution function/degree of belief/whatever with which to label a situation”, any more than there is of “the correct song with which to label a situation”; correct with respect to what criteria?, we must ask. And, clearly, different criteria will pull us in different directions (and may not have any or unique solutions, but all of this is ok, unless we actually had some reason to believe the criteria should uniquely determine a solution).
I’ve never understood how this is supposed to be paradoxical.
I have a fair coin. The rules are, I will show you the result if it is tails always. If it is heads, I flip again and show you the result, whatever it is. What are the odds it is tails that you see?
H-
TH
TT
1:2
In this “paradox”, the first question is about the odds of a fair coin; the second question is about the distribution of waking events.
Probability of tails on first flip: 1/2
Probability of tails on second flip, given a heads a first flip: 1/4
Probability of heads on second flip, given a heads on first: 1/4
Odds of tails versus heads: (1/2+1/4):(1/4) = 3:1
Absolutely. But are you confidant that someone couldn’t point out criteria that you hadn’t thought of, but which, once pointed out, you would agree that you care about? And is it so implausible that such criteria would determine a unique answer to this specific problem, though perhaps not all such problems?
ETA: In other words, it seems that your answer is, “I haven’t figured whether there are criteria that I care about which determine an answer.”