EPTBayesianism and Sleeping Beauty - Page 6
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sam!zdat
United States5559 Posts
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sam!zdat
United States5559 Posts
"it is the case that X" "it is the case that when I perform an utterance claiming X I will be correct" and the probabilities of these two things are different. somebody get me an aspirin  | ||
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hypercube
Hungary2735 Posts
On January 05 2013 05:46 sam!zdat wrote: so on sunday when the philosopher explains sleeping beauty will b e correct if she says that 'there is a 1/3 probability that the coin will have been tails'? She would be correct to say: "When you awaken me you will have thrown tails with a probability of 1/3." She might add: "However when you do not awaken me the probability of you having thrown tails is 1. You will wake me up with a probability of 3/4 and not wake me up with a probability of 1/4. This leads to a total probability of throwing tails as (3/4)*(1/3) + (1/4)*1 = 1/2" Think of it this way: I throw coins and record the results on separate pieces of paper. Now the probability for any particular piece containing the text "Tails" is 50%. Now I announce that I'm throwing out half of the pieces that have "Tails" written on them. Have I changed the probability of the pieces that I didn't touch? + Show Spoiler + The answer of course is yes, now each paper is less likely to have Tails written on them. Even the ones I didn't touch. The key is that I changed the underlying distribution. Saying: "The probability that this piece of paper has Tails written on it is 50%" is imprecise. The precise way to say it is that the piece of paper is part of a distribution that is 50% heads 50% tails. The probability isn't intrinsic to the piece of paper, it's a consequence of it being part of a particular distribution. If I change the distribution I change the probability. | ||
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hypercube
Hungary2735 Posts
On January 05 2013 06:27 sam!zdat wrote: See what's curious about this problem is we seem to have two different claims: "it is the case that X" "it is the case that when I perform an utterance claiming X I will be correct" and the probabilities of these two things are different. somebody get me an aspirin You insist in saying P(tails) and P(tails|waken up) are the same thing. They are not. Of course the probabilities of the two things are different because you changed the meaning of X. | ||
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farvacola
United States18864 Posts
On January 05 2013 06:27 sam!zdat wrote: See what's curious about this problem is we seem to have two different claims: "it is the case that X" "it is the case that when I perform an utterance claiming X I will be correct" and the probabilities of these two things are different. somebody get me an aspirin I smell the despotic obfuscation of truth-in-performativity abound...... | ||
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EtherealDeath
United States8366 Posts
On January 05 2013 00:28 hypercube wrote: What you're looking for is P(Tails|Waken up) By Bayes theorem P(Tails|Waken up) = P(Waken up|Tails)*P(Tails, a priori)/P(Waken up) = 0.5*0.5/0.75= 1/3 I could go back and check your math, but in these cases the Bayes theorem always gives the same result as the traditional way of counting elementary cases. If you get a different result you messed up somewhere since the two ways are mathematically equivalent. edit: Which is exactly the same as you got, so I don't understand the problem. I was calculating the probability that it is Monday, given that we are sitting down for Tea, not the probability that the coin is tails given that we woke up. | ||
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hypercube
Hungary2735 Posts
On January 05 2013 06:43 EtherealDeath wrote: I was calculating the probability that it is Monday, given that we are sitting down for Tea, not the probability that the coin is tails given that we woke up. Yea, I got that later. Anyway, I couldn't find your calculation for P(Tails|Tea). Do you have any issues with mine? | ||
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sam!zdat
United States5559 Posts
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imallinson
United Kingdom3482 Posts
On January 04 2013 14:51 sam!zdat wrote: Then the philosopher asks: Sleeping Beauty, my ravishing somnolent darling, what credence do you ascribe to the proposition that "the Coin was tails" That's the important point. While you could make an argument that a question about whether it was Monday or not could be deemed to be a 1/3 probability that isn't the question. Sleeping Beauty's awakeness has no effect on what the coin ended up so the probability is 1/2. edit: Thinking about it more I don't think you could argue the Monday being 1/3 thing either. Yes there are three times she wakes up but P(Heads|Monday) and P(Heads| Tuesday) are the same result so they both have probability of 1/2. edit: Ok I'm an idiot the Monday thing is 1/3. Point still stands about the probability. | ||
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farvacola
United States18864 Posts
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sam!zdat
United States5559 Posts
On January 05 2013 06:29 hypercube wrote: She would be correct to say: "When you awaken me you will have thrown tails with a probability of 1/3." So what is the difference in the formal structure of claim "you will have thrown tails" and "when you awaken me you will have thrown tails"? Then when she actually wakes up and considers the claim "you have thrown tails" should she believe 1/3 or 1/2? Remember that she didn't gain any information about the world, and before she went to sleep the probability was 1/2. She might add: "However when you do not awaken me the probability of you having thrown tails is 1. You will wake me up with a probability of 3/4 and not wake me up with a probability of 1/4. This leads to a total probability of throwing tails as (3/4)*(1/3) + (1/4)*1 = 1/2" Yes, so that's how you know on Sunday that the probability is 1/2. But when she wakes up she doesn't know what day it is. What should she answer about her belief that the coin was tails? Think of it this way: I throw coins and record the results on separate pieces of paper. Now the probability for any particular piece containing the text "Tails" is 50%. Now I announce that I'm throwing out half of the pieces that have "Tails" written on them. Have I changed the probability of the pieces that I didn't touch? + Show Spoiler + The answer of course is yes, now each paper is less likely to have Tails written on them. Even the ones I didn't touch. The key is that I changed the underlying distribution. Saying: "The probability that this piece of paper has Tails written on it is 50%" is imprecise. The precise way to say it is that the piece of paper is part of a distribution that is 50% heads 50% tails. The probability isn't intrinsic to the piece of paper, it's a consequence of it being part of a particular distribution. If I change the distribution I change the probability. you're assuming that distributing items in a set in space is the same as distributing them in time. Is that true? | ||
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EtherealDeath
United States8366 Posts
On January 05 2013 06:45 hypercube wrote: Yea, I got that later. Anyway, I couldn't find your calculation for P(Tails|Tea). Do you have any issues with mine? Hmm I suppose that could work actually. | ||
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hypercube
Hungary2735 Posts
On January 05 2013 06:46 sam!zdat wrote: Why are you guys talking about tea? The tea is utterly irrelevant I just put it in there because it sounds like something Lewis Carroll would write. Have you read what I wrote? | ||
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sam!zdat
United States5559 Posts
On January 05 2013 06:46 imallinson wrote: That's the important point. While you could make an argument that a question about whether it was Monday or not could be deemed to be a 1/3 probability that isn't the question. Sleeping Beauty's awakeness has no effect on what the coin ended up so the probability is 1/2. yes, good. But then how do you explain the EV? The fact remains that she'll probably be wrong if she says "the coin was tails" | ||
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sam!zdat
United States5559 Posts
oh, you mean "tea" to signify "I'm currently awake"? | ||
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sam!zdat
United States5559 Posts
On January 05 2013 03:10 hypercube wrote: Don't see how you can say that "no new information was gained". The fact that she was woken up IS new information. What information? She knows "it is now monday or tuesday, and no longer sunday." That is an indexical. But she doesn't learn anything about the coin because she has the same exact experience no matter what. She could have predicted that this would happen on Sunday, when she believed unproblematically that the answer was 1/2, so I don't see how that gains her information. edit: hypercube can you summarize your position, sorry I'm trying to figure out what you're answering but I'm confused a bit. | ||
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imallinson
United Kingdom3482 Posts
On January 05 2013 06:51 sam!zdat wrote: yes, good. But then how do you explain the EV? The fact remains that she'll probably be wrong if she says "the coin was tails" I think I've been thinking about this wrong. It's not whether its heads or tails that's really being asked. edit: For an outside observer its definitely still 50-50. Not so sure from her perspective. | ||
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sam!zdat
United States5559 Posts
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Boblion
France8043 Posts
On January 05 2013 06:18 Kukaracha wrote: But would you deny that we make many, many educated guesses in real-life settings? Poker, stock exchange... It's true that the coin example would be flawed IRL because it can't be calculated accurately enough, but my questions remain. I have never denied the usefulness of maths lol. Obviously we have to use this knowledge to make guesses in real-life settings but we are making assumptions, we are theorizing. Again do not mistake the maths with the "real" stuff. Everyone with basic probabilities knowledge can understand the "maths" behind poker (and make the assumption that cards are dealt randomly) on the other hand if you know and understand perfectly the Pseudorandom number generator of a poker site you gonna be rich real quick. There are more things than just the "maths" in poker, you can also get some information by observing the other players (or their cards lol). Knowing the maths will help you tho and on the long run you should be able to beat a guy who have no idea about the basic strenght of the hands. Or you could get unlucky. Stock exchange is way more complicated and i don't really want to discuss this but it is definitly not just about maths (or you could argue that our maths models are not strong enough yet) and there are way too many things involved. Insider trading will make you richer than having 300 IQ and a Fields medal. Or you could go to jail lol. But let's just say that it is like a giant poker game with millions of players and a crazy amount of cheating and randomness. | ||
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imallinson
United Kingdom3482 Posts
On January 05 2013 07:08 sam!zdat wrote: Ah, good! what is really being asked? It's a question of whether it's heads or tails as a function of what day it is. That made like zero sense. There are two possibilities for what day it is. Monday can only occur if it's heads in which case it is 100% heads. Tuesday can occur if it's heads or tails so you have double the chance of getting Tuesday. If it's Tuesday its 50-50 heads or tails. P(Mon|Heads)=1/3 x 1 = 1/3 P(Tues|Heads)=2/3 x 1/2 = 1/3 P(Tues|Tails)=2/3 x 1/2 = 1/3 So P(Heads) = P(Mon|Heads) + P(Tues|Heads) = 2/3 and P(Tues) = 1/3 So go with tails. | ||
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