• Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    If you toss a fair coin 100 times and throw one red marble into an urn on heads and two blues on tails, you'll end up with (roughly) 50 reds and 100 blues. If you count each of the marbles as an outcome of the coin toss, without discounting the blues, you'll end up with 100 tosses having 150 outcomes, which is absurd. It's an attempt at alchemy.

    One of the side effects of this attempted alchemy is that each red represents twice as much of an outcome as each blue. Yes, there is something absurd about the 2/3, but it's a result of putting in twice as many blues per toss but then taking them out one at a time, as if they were the same as the reds. You can pair off each red with two blues -- that is, taking the marbles back out of the urn the same way you put them in -- without absurdity. If you insist on taking the blues back out singly, the absurdity of the result (a marble representing half an outcome) is on you.

    As for Beauty's state, try thinking of the Tuesday interview this way: I am (still) being interviewed about a tails outcome (the same one as yesterday). There was just the one coin toss, with just the one outcome. Smearing the interview across two days doesn't change that. Beauty does not know which interview this is, but she knows there will be two interviews for each tails outcome and she discounts accordingly.

    Think about what's going on if she wagers. She can make a Dutch book on tosses of a fair coin at even money. That should not be possible. That's just as strong a principle as the business about no updating without new information. If it is possible, someone's performing alchemy or cheating.
  • Mathematical Conundrum or Not? Number Five
    Discounting doesn't help Beauty. Suppose she is a halfer. Should she discount the next drawn marble (or current interview) by 1/2 or not? Well, she should 2/3 of the time. So 2/3 * (1/2 * 1/2) + 1/3 * 1/2 = 1/3. The thirder says that is the real probability of each state for her. — Andrew M

    This is the wrong model. This table
        Mon   Tue
H   1/2
T   1/4   1/4
    is right, and here's why.

    Suppose you have a machine set up like this: there's a hopper full of red marbles and a hopper with twice as many blue marbles; you push one button and it transfers a single red marble or two blue marbles to another hopper, one you can't see; you push a different button and it dispenses one of the marbles from the small hopper. What are your odds of getting a red marble? 1/2. Half the time only a single red marble goes into the small hopper and then gets dispensed in the second step. (Half the time, two blue marbles go in, and then one of those two is dispensed, so the chance of blue -- a blue, any blue, one of the two in the small hopper -- is also 1/2, despite the fact that twice as many marbles were dispensed at the stage you don't see.)

    Now do it this way: you have a hopper full of white marbles; push one button and half the time a single marble is moved to the small hopper, half the time two; you push the second button and get a single marble. How many marbles are left in the small hopper? Dunno. Half the time there's still one there, and half the time there isn't.

    You could randomize. Any number of marbles could be dispensed to the small hopper. Getting one tells you exactly nothing. You could have it transfer a random number of reds to the small hopper half the time and a random number of blues half the time. When you push the second button to get your marble, the chances will still be 50:50 of getting a red or a blue.
  • To See Everything Just As It Is
    We love the things we love for what they are.
  • Mathematical Conundrum or Not? Number Five
    @Jeremiah
    It's as we were discussing: each marble represents an interview event. To count coin toss outcomes you only need one red marble, but two blues, to make up the entire double interview event. Each blue marble is one kind of event, but that event is half of the kind of event we want to count.
  • Mathematical Conundrum or Not? Number Five
    @Michael I think I'm a halfer now. (Still some things I'd like to be clearer on.)

    @Andrew M, @andrewk, @JeffJo: do you find this argument as convincing as I do?

    @Jeremiah
    each blue is discounted, and is only half the total available evidence of a tails flip, unlike the reds each of which is all the evidence of a heads — Srap Tasmaner

    Is this what the conditional probabilities we've been talking about are trying to express? I'm still not clear on how this idea is formalized.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Something I don't remember us talking about: should Beauty, knowing the rules of the experiment, subject her expectation of a tails interview to a discount? It occurs to me that this may be the regime Lewis is describing.

    Here's a physical version. You decide to test if a coin is fair by throwing a red marble in an urn on heads, and a blue marble if tails. After a bunch of flips, you'll count the marbles, expecting them to be about equal. Drawing a marble randomly will have the same distribution as the coin itself.

    Suppose instead on tails you throw in two blue marbles. Then you'd expect a 2:1 ratio if the coin is fair. A randomly selected marble is now twice as likely to be blue, but each blue is discounted, and is only half the total available evidence of a tails flip, unlike the reds each of which is all the evidence of a heads. Each blue does represent a tails, certainly, and only got in the urn because a tails was tossed, but there's another blue that's evidence of the same toss.

    Now suppose the marbles are all white. Still true that you're twice as likely to draw a marble representing a tails toss, but you have to discount.
  • Mathematical Conundrum or Not? Number Five
    The conditional probability of tails given that it is M2 or Tu is P(T|M2)= P(.5|.25) = .125/.25 = 1/2 = P(T|Tu) which is equal to P(H). — Jeremiah

    This is still slightly puzzling to me.

    P(H | M1) = 1, right? And this is the thing about the double interview track: both them happen if and only if the coin lands tails. From your calculation, P(T | M2 v Tu) = 1, yes? But it should be P(T | M2) = P(T | Tu) = 1, and P(T) = P(M2) = P(Tu) = 1/2. You always get both on tails. You get them one at a time, but we don't necessarily care.

    That space of three possibilities, {M1, M2, Tu} has three elements each of which has an unconditional probability of 50%. Conditioned on the whole space, they'll each be 33%.
  • Mathematical Conundrum or Not? Number Five
    In the event of Tails, Beauty will be awakened on Monday and Tuesday, but due to the nature of the experiment she will not be able to tell the difference, either one is equally likely when interviewed — Jeremiah

    So what? It's not a situation that arises. Neither she not the experimenters are ever in the position of knowing that the coin landed tails but wondering what day it is. Beauty only wonders what day it is to figure out how the coin landed.


    Suppose there was another coin toss to determine whether heads was the single interview or the double this time around. Then half the time heads would be 1/3 of the interviews, and half the time 2/3, so heads would on average be half the interviews and same for tails.

    But that is not the case here. The interviews are not randomly distributed.

    Beauty knows that when she is asked for her credence, 1/3 of the time the coin has landed (or will land?) heads and 2/3 of the time the coin has landed (or will land?) tails.

    Therefore her credence that the coin has landed (or will land?) heads must be 1/3.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    I misunderstood.

    But you are going to say exactly this about M2 and Tu (or T1 and T2), so the question stands.
  • Mathematical Conundrum or Not? Number Five
    H has one element and set T has two elements, each with a 50% chance of being selected. — Jeremiah

    When does anyone ever make a random selection from among only the tails interviews?
  • Mathematical Conundrum or Not? Number Five
    Continuing:

    How many times does Beauty expect to be asked for her credence?
    (1) If I knew it was heads, I'd know I'll be asked once.
    (2) If I knew it was tails, I'd know I'll be asked twice.
    (3) It's heads half the time and tails half the time.
    (4) Therefore my weighted expectation is 1/2(1) + 1/2(2) = 3/2.

    If I'm in the single interview track, and I am half the time, I get 2/3 of my expected interviews.
    If I'm in the double interview track, and I am half the time, I get 4/3 of my expected interviews.

    My expectation for getting to say "heads" and be right, because I'm in the single interview track, is 2/3(50%) = 1/3.
    My expectation for getting to say "tails" and be right, because I'm in the double interview track, is 4/3(50%) = 2/3.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Thanks. I think I finally understand the halfer position. (The one thing I'm not completely clear on is how the Monday interview is retroactively determined to be a single or half of a double in the variant where the coin is tossed after the first interview.)

    What puzzles me is why Beauty would reason this way.

    My Beauty reasons this way, as I've said before:
    (1) If I knew it was Monday, I'd know it could be heads or tails, even chance.
    (2) If I knew it was Tuesday, I'd know it was tails.
    (3) I know I'll be interviewed on Monday, but interviewed on Tuesday only half the time.
    (4) Therefore my weighted expectation of heads is 2/3(1/2) + 1/3(0) = 1/3

    The halfer Beauty reasons this way:
    (1) If I knew I was in the single interview track, I'd know it was heads.
    (2) If I knew I was in the double interview track, I'd know it was tails.
    (3) I'm in the first track half the time and in the second half the time.
    (4) Therefore my weighted expectation is 1/2(1) + 1/2(0) = 1/2

    But this is just pretend reasoning.

    It's like "working out" your expectation of heads in a simple coin toss this way:
    (1) If I knew it was heads, I'd know it was heads.
    (2) If I knew it was tails, I'd know it was tails.
    (3) It's heads half the time and tails half the time.
    (4) Therefore my weighted expectation is 1/2(1) + 1/2(0) = 1/2

    What's the point of that?

    And indeed, Lewis's "proof" has but a single step.

    (No argument in this post, just clearing my head.)
  • Mathematical Conundrum or Not? Number Five
    UNIMPORTANT ASIDE:

    I think this problem arose out of earlier problems and chitchat about decision-making given imperfect memory.

    I keep thinking that if it had developed on its own, it would be a time travel puzzle. "Tuesday" appears here essentially as "another Monday". You have no way of knowing for sure it's the first Monday or the last you will experience, etc. etc.
  • Mathematical Conundrum or Not? Number Five
    When you see Lewis assign 1/4 to T1 and 1/4 to T2 then what are looking at is the frequency in which is he assigning to those two possible outcomes from the 1/2 in the event of tails. He saying that since being awakened on Monday and Tails and Tuesday and Tails is under the same event then you are equally likely to be in either one upon being awakened and interviewed. However, that does not mean they are pulled from a separate sample space. You have to understand that Monday Tails and Tuesday Tails are pulled from the same event of the coin landing on tails, which is 1/2. — Jeremiah

    The double interview is not a single event, for the simple reason that Beauty makes two decisions.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    I know what you meant by "{H,T}". I was asking what your point is.

    You know the math better than I do, so if you have something to say, I'm going to listen.

    I'm willing to do the work to figure things out on my own, but what you have in mind is not one of those things.

    Can we just get back to SB now?
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Meaning?
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    Here's the picture halfers actually use:

    0fqxqKd.jpg

    And I think they use the same sort of weighted expectation I keep posting, only theirs looks like this:





    There are a couple of curiosities here:
    • in the first (and perhaps only) interview, heads is twice as likely as tails;
    • there aren't half as many second interviews as firsts, but one third.
    I find these proportions strange. Lewis ends up here and shrugs. I'm not sure what to make of it, but this is a far cry from the way I think halfers want to think of their position, that it's just a coin flip with some meaningless frosting on it. Whichever position we take, something about it is counterintuitive.
  • Mathematical Conundrum or Not? Number Five
    when you get right down to it: it is still just a coin flip. — Jeremiah

    And when I see the word "just" used as it is here, I assume someone is trying to manipulate my intuitions. If ever a coin flip wasn't "just" a coin a flip, surely it's Sleeping Beauty.

    I'm for trying to tackle the problem as posed. I don't think we should assume, for instance, that Beauty is informed by being awakened. But I'm also for examining the problem statement carefully to see if information is being smuggled to Beauty.

    Assume you need information to update your prior; either Beauty doesn't update (and the problem has manipulated our intuitions to suggest she has) or she has received information (and the problem makes it appear she hasn't) or some third option (like a really complicated credence function).

    We don't know which. Maybe the problem is just underdetermined or ambiguous (but is pretending it can be answered). We have to look. Eliminate the blind alleys. Maybe when we're done there will be a choice we can't eliminate, maybe not.
  • Mathematical Conundrum or Not? Number Five
    ↪Andrew M

    The post I referenced had a mistake!

    ($2 bets below for simplicity, since the coin is fair.)
    Before I gave the SB payoffs at even money as
             Bet
         H   T
Toss H   1  -1
     T  -1   2
    and noted that heads will break even while tails makes a profit. That's wrong. The right table is obviously
             Bet
         H   T
Toss H   1  -1
     T  -2   2
    because you bet heads incorrectly twice when the coin lands tails.

    Thus the SB 2:1 table would be:
             Bet
         H   T
Toss H   2  -1
     T  -2   1
    and everyone breaks even. 2:1 are the true odds.

    For a reminder, the single toss for a 3:1 biased coin ($2 bet for consistency):
              Bet
          H    T
Toss H    .5  -.5
     T  -1.5  1.5
    Same as the SB results: heads loses $1, and tails earns $1.

    And, no, obviously SB doesn't break even on 3:1 bets:
             Bet
         H   T
Toss H   3  -1
     T  -2   2/3
    At odds greater than 2:1, heads will be the better bet.

    Sleeping Beauty remains its own thing: the odds really are 2:1, but the payoffs are 3:1.

    (Disclaimer: I think this is the most natural way to imagine wagering, but you can come up with schemes that will support the halfer position too. They look tendentious to me, but it's arguable.)
  • Mathematical Conundrum or Not? Number Five
    quarterer — Andrew M

    Mainly so we'd get to use that word.

    This is all stuff we've said before -- this comment summarizes the mechanism by which standard thirder wagering pays out 3:1, as @andrewk pointed out, instead of 2:1.

    You could also think of it as revenge against the halfer position, which draws the table this way:

    SSiqXYS.jpg

    Halfers, reasoning from the coin toss, allow Monday-Heads to "swallow" Tuesday-Heads.

    Reasoning from the interview instead, why can't we do the same?

    plbhebN.jpg
  • Mathematical Conundrum or Not? Number Five
    Yet another take 1. (More to come.)

    Ignore the coin toss completely. The intention of the problem is that Beauty cannot know whether this is her first or second interview. If we count that as a toss-up, then





    That is, Beauty would expect a wager at even money to pay out as if there were a single interview and the coin was biased 3:1 tails:heads. And it does.
  • Ongoing Tractatus Logico-Philosophicus reading group.
    ↪Posty McPostface

    I remember reading at least some of Pears's The False Prison years ago and was impressed (v.1 is early LW, v.2 late). He argues for lots of continuity as I recall, so that's interesting.

    ((I was back then too enamored of the later stuff to study TLP seriously...)
  • Mathematical Conundrum or Not? Number Five
    We can then distinguish between the case of telling someone twice that it's tails and the case of telling someone once but using a weighted coin that favours tails. — Michael

    The problem with SB is that the outcomes are like a 2:1 biased coin, but the payouts (as @andrewk pointed out) are like a 3:1. If we ignore wagering, could SB tell the difference between the official rules and a variant with a single interview and a biased coin? If she can't, is that an argument in favor of one position or the other?

    From the other side, wagering will tell my SB that it's not a biased coin but a bizarre interview scheme. Will a halfer SB be able to tell the difference?
  • Mathematical Conundrum or Not? Number Five
    the proper way — Michael

    Sleeping Beauty is a pretty unusual situation though.

    Some of us think it merits switching to counting occasions instead of counting classes of occasions. There are two ways to do it. YMMV.

    On our side there are confirming arguments from wagering and weighted expectations. On the halfer side I only see the "no new information" argument.
  • Mathematical Conundrum or Not? Number Five
    ↪Michael

    There's a 50% chance you'll tell me "at all" that it's heads, and same for tails. But there's more than a 50% chance that a random selection from the tellings you've done will be a tails telling.
  • Mathematical Conundrum or Not? Number Five
    the Bayesian approach — Jeremiah

    If that means the "subjective" interpretation of probability, it's just what the question is about.

    Maybe it ends up showing that "degree of belief" or "subjective probability" is an incoherent concept and we all become frequentists.
  • Mathematical Conundrum or Not? Number Five
    ↪Michael

    Is it?

    I think the halfer intuition is that a coin toss is a coin toss -- doesn't matter if you're asked once on heads and twice on tails.

    But consider this. What is your expectation that I'll tell you it was heads, given that it was heads? 100%. What's your expectation that I'll tell you it was tails, given that it was tails? 100%. Does that mean they're equally likely? To answer that question, you have to ask this question: if I randomly select an outcome-telling from all the heads-tellings and all the tails-tellings, are selecting a heads-telling and selecting a tails-telling equally likely? Not if there are twice as many tails-tellings.

    Both conditionals are certainties, but one is still more likely than the other, in this specific sense.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Done a little more sniffing around, and thirders frequently argue there's information here. Elga doesn't. <shrug>

    As SB, you are asked for your degree of belief that a random event has occurred or will occur.

    If I flip a fair coin and ask you for your degree of belief that it landed heads, you'll answer 50%.

    Suppose instead I say I'm going to tell you how it landed. What is your degree of belief that I'm going to tell you it landed heads? It will again be 50%. They're usually identical.

    Now try this with SB: instead of asking for your degree of belief, I'm going to tell you how the coin toss landed. What is your degree of belief that I will tell you it landed heads? Is it 50%?

    We thirders think halfers are looking at the wrong event. Just because you're asked how the coin landed, doesn't mean that's the event you have to look at to give the best answer.

    (I've also got a variation where I roll a fair die after the coin toss, and ask or tell you twice as frequently on tails. Same deal: what's the random event? Is it just the coin toss?)
  • Mathematical Conundrum or Not? Number Five
    However, she will be able, and she will be taught how, to distinguish her brief awakenings during the experiment from her Wednesday awakening after the experiment is over, and indeed from all other actual awakenings there have ever been, or ever will be. — Lewis

    Hey look at that. He saw the Wednesday argument and slipped in a defeater!
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Huh. Didn't realize my first argument might be a contradiction.

    I'll slog through the Lewis some more. He also notes that you can't jump straight to indifference about which of the three interviews is happening -- although you can argue for them being equiprobable -- and I didn't quite get that either.

    If your position is that it's just a prior and you can pick whatever you want (thus Bayesianism is intellectually bankrupt), then the section at the end of Lewis's article is relevant right? It was all this stuff about believing now a credence you know you should believe in the future.

    Is there any argument for not to reporting both the 1/2 and 1/3? Seems like a perfectly valid solution to me and if I ever has a like dilemma in the real world that is exactly what I would do. — Jeremiah

    I think "What do you mean?" is a good answer. "If you weren't asking me in this bizarre manipulative way, I'd say 50%; since you are I'll say 33%."
  • Mathematical Conundrum or Not? Number Five
    The simulation proves that both views are valid ways to look that the possible outcomes. — Jeremiah

    I understand Elga's argument. I understand the wagering argument. Do you understand Lewis's argument? I don't. He tries to get you to accept P+(HEADS)=2/3 because of something about prophecy and backwards-in-time causation and accepting credences you wouldn't normally. I've read it several times but I don't know what he's on about.
  • Mathematical Conundrum or Not? Number Five
    Oh right. Give the 50% answer you'd give on Monday, because there are more Monday interviews. I remember thinking about that a while ago -- you get to be right 2/3 of the time. — Srap Tasmaner

    This no good though. If you know you're wrong 1/3 of the time, your credence is really the 33%.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    Oh right. Give the 50% answer you'd give on Monday, because there are more Monday interviews. I remember thinking about that a while ago -- you get to be right 2/3 of the time.
  • Mathematical Conundrum or Not? Number Five
    Geez, I've been staring at this far too long. The magic 1/6 is right there. We already have credence that it's Monday, when our credence would be 50%. Tell me it's Monday and I get to add the last , or 1/6.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah

    The simplest way to block the Wednesday argument is to change the experiment: they send you on your way immediately after your last interview. It's cleaner. The wake you up on Monday, flip a coin, and ask for your credence. If it was heads, they send you on your way. If it was tails, you get the amnesia sleep, then they wake you up on Tuesday, ask for your credence and send you on your way. No letting you sleep through Tuesday, on heads, no waking up Wednesday not knowing if you'll be interviewed.

    Now when you wake up, all you have is the original weighted average:



    I don't know if anything here counts as "updating".

    One might argue that all things being equal Beauty should go with the 50% credence, since 50 is greater than 33, which gives her a higher chance of being right. If the both argument are equal that is. — Jeremiah

    The problem is that wagering confirms the odds are 2-1, which, duh, there are 2 tails interviews for every heads interview. If it's all about getting to give the right answer most often, there's no way to go but tails.
  • Mathematical Conundrum or Not? Number Five
    The problem with Elga's argument is that can't count your temporal location relevant if you don't know what it is. — Jeremiah

    When you are first awakened, you are here:



    The 's there are the days -- you don't know if it's Monday or Tuesday or Wednesday, but you know you've been awakened.

    (Do we really have to say you already knew this was going to happen? That you were certain? What if you're awakened by a bloody research assistant surrounded by rubble? Do you assume the experiment is still running? What's become of this certainty about the future Elga attributes to you?)

    As soon as you're asked for your credence, you switch to



    and now you're weighting by possible interviews per day.

    Wednesday is not from the same population of awakening where she is interviewed. — Jeremiah

    Well, yeah. She's awakened but not interviewed. According to the setup this does happen.
  • Mathematical Conundrum or Not? Number Five
    1/2+1/6 = 2/3

    P+(Heads) = P(Heads)+1/6     — Jeremiah

    That is P(HEADS | you told me it's Monday) = P(HEADS | you asked me) + 1/6

    For Elga, that's 1/2 = 1/3 + 1/6

    Here's how I got the idea to include Wednesday, and it's still a good argument, I just dropped it because it all comes down to counting anyway.

    Lewis accepts that being told it's Monday, and therefore not Tuesday, is relevant new information. (It tells you you're not in the future!) He does not accept that being asked for your credence is relevant new information, but it is: it tells you it's not Wednesday. If he accepts one, he ought to accept the other. Game over.

    I had this whole analysis worked out about Beauty's credence upon being awakened Monday or Tuesday or Wednesday before being interviewed -- awakened on Wednesday, she won't know the experiment's over until they tell her. You can redo the weighted average credence I did before (and which is the same argument Elga makes) including Wednesday:



    I still find that convincing. Maybe it's even more convincing than



    because it bakes in why being asked the question is informative, and why Lewis ought to be willing to conditionalize on H1 ∨ T1 ∨ T2.
  • Mathematical Conundrum or Not? Number Five
    ↪Jeremiah
    No worries. The only reason to throw in awake-but-unasked is to show there's yet another way to carve up Beauty's credence. (And we cannot leave her asleep forever -- part of the setup is awakening her and sending her on her way. She will know when this has happened.)

    1/6 is just 1/2 - 1/3, of no interest in itself, and we don't really need to throw in the Wednesday states or worry about whether the experiment concludes only on Wednesday or on either of Tuesday or Wednesday. Some denominators change. None of it matters. It's still just counting.

    There's no conflict between believing P(HEADS) = 1/2 unconditionally and believing P(HEADS | you're asking) = 1/3. That's all there is to it. If asked, she should say "1/3". I don't think there's anything like her credence changing going on it at all. It only looks that way because she's answering the question "What is your credence?" If you don't ask, it's one thing; if you do, it's another. Just the usual silliness.
  • Mathematical Conundrum or Not? Number Five
    No that's wrong.

    We can imagine that on HEADS, we wake up Beauty Tuesday and send her on her way. That's no different from putting her back to sleep after a Tuesday interview and waking her up Wednesday. She will not know the difference between sleeping two days and only one. So that would be a sample space of 5 instead of 6 -- the experiment can conclude on two different days, depending on the coin, so that's not the explanation for the magic 1/6.

    It's also the same as not counting Tuesday & Heads.
  • Mathematical Conundrum or Not? Number Five
    Tuesdays and Heads is clearly and obviously not in the sample space. It is not a possible outcome, she will be never be interviewed on Heads and Tuesday. — Jeremiah

    It is a possible outcome; she just won't be asked about it.

    What's more, Wednesday & Heads and Wednesday & Tails are both part of the sample space. In those cases, she'll be awakened but not asked.

    We can only ask her about her credence if she's awake; and we can't ask her if we're not asking her. But she can have a credence whether she's awake and whether we ask.

    Think about this -- the variation we haven't discussed -- why does Beauty's credence in HEADS increase by 1/6 when she's told it's Monday? For Elga, it's an increase from 1/3 to 1/2 (and he's right), but for Lewis it's from 1/2 to 2/3 (and he's wrong). Where does this magic 1/6 come from?

    It comes from the sample space, which has 6 equiprobable possibilities.

    We can ask what Beauty's credence that a fair coin will land HEADS:
    • whether we awaken her or not: 3/6
    • given we've awakened her: 2/5
    • given that we're asking: 1/3
    • given that we told her it's Monday: 1/2
    • given that it's Tuesday: 1/2
    • given that it's Tuesday and she's awake: 0
    • given that it's Tuesday and we're asking: 0.

    Looked at this way, it's a whole lot of fuss over nothing. The whole point of all the epistemic this-and-that, the centered and uncentered worlds business, all that, is just to get us to pretend we don't know things we know, on the theory that Beauty could not possibly know them, that she is in some astonishing epistemic quandary.

    Beauty just needs to know how to count. Her credences are all consistent given the various conditions we might impose. It's pure semantics really to say that her credence changes at all.
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