Sleeping Beauty Paradox

[KipBond]

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"What is the probability that you were awakened by a tails-flip?" the answer is 2/3.
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Since the amount of "heads flips" and "tails flips" are equivalent, she will state that the probability of the coin being flipped heads or tails is 50% for each.

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So, after being awakened, what is her answer to the question: "What is the probability that the coin was heads?"

The question is ambiguous. You give answers to both interpretations above.

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There is no ambiguity in the question. The answer to the question, "What is the probability that the coin was heads?" is 1/2. The answer to the question, "What is the probability that you were awakened this time by a 'tails flip'?" is 2/3.

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That doesn't make sense. I assume you would say the answer to "What is the probability that you were awakened this time by a 'heads flip'"? is 1/3? When I say the coin was heads that can mean the coin was heads when I flipped it, or the coin was heads this time prior to me waking you up. They are different.

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These questions are not the same. The first refers to the probability of the state of the coin, while the second refers to the probability that she was awakened by a specific flip. The probability of the state of the coin and the probability that she was awakened by a specific flip are not the same.

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Yeah, I think I agree with you here. The questions are not the same, but the original question can be interpreted two different ways, resulting in 2 different answers. Thus, the ambiguity.


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Since different types of flips result in different numbers of awakenings, but flipping the coin results in an equal number of heads and tails, it follows that the probabilities for each question are different. However, neither question is ambiguous; both have definitive answers.

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It's ambiguous as to which question is being asked. Ambiguity is obviously subjective.

[PairTheBoard]

Suppose you Flip and put 1 black ball in the bag if heads is flipped. And you put 100 White balls in the bag if tails is flipped. Now pretend you are a ball in the bag. You can't tell your color nor how many balls are in the bag. You are asked, what is the probability you are a black ball?

Shouldn't you say there is a 50% probability you are a black ball? What does that probability mean? It means that if we repeat this experiment you will find yourself being a black ball in half the experiments. You will find yourself being a white ball in half the experiments. And it the white balls are numbered 1-100 you will find yourself being White Ball #100 in .5% of the experiments.

Beauty's awakenings are exactly the same as the balls in the bag.

PairTheBoard

[PairTheBoard]

Here's the thing. When you give a probability you are giving a frequency wrt "number of times". Our probability "number of times" here is "number of times" the experiment is run. The 1/3 people are changing the probaility to "number of times" the awakening happens. Do you want the frequency per Experiment, or do you want the frequency per Awakening?

When Beauty is asked, what's the chance this is Monday, does she want to be right per experiment or per Awakening?

Consider my 1 Black Heads ball or 100 White Tails balls in the bag example. The white balls are numbered 1-100. You are a ball in the bag. When Tails, you will be White Ball-1 on day 1, White Ball-2 on day 2, ..., White Ball-100 on day 100. When Heads you will be the Black ball on day 1 after which you will be removed. You will be asked this question on day 1. When Tails you will be asked this question on each of days 2-100.

You are a ball. What is the probability you are a Black Ball? If you answer 50% you will be right per experiment. ie. In half of the experiments, when you are asked that question you will be a black ball.

If you answer 1/101 you will be right per Ball experience. In 1 out of 101 ball experiences you will be a Black Ball.

When Beauty is asked the probability she is having a Heads-Awakening does she want to be right per experiment or per Awakening experince?

If she answers 1/2 she will be right per experiment. In half the experiments, when she is asked that question she will be in a Heads-Awakening.

If she answers 1/3 she will be right per Awakening experience. In 1/3 of her awakening experiences she will be in a Heads-Awakening.

So what should Beauty's probability measure be. The one she started out with, ie. per Experiment. Or a new one she creates for some reason which is per Awakening. Why should she adopt a new probability measure? If she insists on using this new probability measure of her creation why doesn't she just give both probabilites to be clear?

Isn't the full answer just the one Deorum is saying she should give?

She can simply say, the probability I am in a Heads Awakening is 1/2 per Experiment and 1/3 per Awakening.

PairTheBoard

[Deorum]

Kipbond,

Ah, okay I think I understand now what you mean by ambiguity. I think all three of us are pretty much in agreement. Let's recap:

If she is asked, "what is the probability that you were awakened by a 'heads flip' this awakening", her answer will be "1/3" because the total number of heads-awakenings over an average trial period will be 1/3. Likewise, tails-flips awakenings constitute 2/3 of the total awakenings. Are we all in agreement about this?

If she is asked, "what is the probability that the coin came up heads when we flipped it" her answer will be "1/2" because she knows that she is twice as likely to have been awakened at any given point by a tails-flip, as well as knowing that the parameters of the experiment are such that she is awakened twice as often when the coin comes up tails, and thus will conclude that the coin came up tails half as often as she is awakened by tails-flips thereby reducing that number by half. The result of this reduction is that the amount of heads-flips and tails-flips are equal, giving a probability of 1/2 to each. Are we in agreement about this?

So as long as we can agree with these two parameters, there is no paradox, right? It is simply unclear as to which question the OP is asking (although I would argue that the question as posed, "What is the probability that the coin came up heads", infers that they mean the absolute state of the coin, regardless of which caused her to wake, such that the answer is 1/2 - but okay, semantics). It appears that the only problem is in the clarity of the question stated by the OP, not what is actually going on mathematically, correct?

[KipBond]

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Here's the thing. When you give a probability you are giving a frequency wrt "number of times". Our probability "number of times" here is "number of times" the experiment is run. The 1/3 people are changing the probaility to "number of times" the awakening happens. Do you want the frequency per Experiment, or do you want the frequency per Awakening?
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So what should Beauty's probability measure be. The one she started out with, ie. per Experiment. Or a new one she creates for some reason which is per Awakening. Why should she adopt a new probability measure? If she insists on using this new probability measure of her creation why doesn't she just give both probabilites to be clear?

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That's pretty much what I've been saying.(*) But, it doesn't require a "changing" of her probability measure after she wakes up. She's perfectly logical, so given the experiment parameters, she can answer both ways before, during, and after the experiment. Just like we can, right now.

Since there really are two different events (coin-flips, and coin-flip-wakings), it's not clear which one should be the probability space in regards to the question being asked. Answer both ways to remove any ambiguity.

(*) Although, the "number of times" doesn't refer to the "number of times" the experiment is run vs. the "number of times" she is awakened. It refers to the "number of times" the coin is tossed, vs. the "number of times" she is awakened. Both events are in the same experiment; it just so happens that the coin is tossed the same # of times the experiment is run, while the awakenings occur 1.5x more frequently. This could be different with different experiment parameters.

[KipBond]

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If she is asked, "what is the probability that you were awakened by a 'heads flip' this awakening", her answer will be "1/3" because the total number of heads-awakenings over an average trial period will be 1/3. Likewise, tails-flips awakenings constitute 2/3 of the total awakenings. Are we all in agreement about this?

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Agreed.

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If she is asked, "what is the probability that the coin came up heads when we flipped it" her answer will be "1/2" because she knows that she is twice as likely to have been awakened at any given point by a tails-flip, as well as knowing that the parameters of the experiment are such that she is awakened twice as often when the coin comes up tails, and thus will conclude that the coin came up tails half as often as she is awakened by tails-flips thereby reducing that number by half. The result of this reduction is that the amount of heads-flips and tails-flips are equal, giving a probability of 1/2 to each. Are we in agreement about this?

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Agreed. Although if she is being asked about the outcome of the coin flip event, then she can just disregard the entire experiment and answer as she normally would: 1/2.

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So as long as we can agree with these two parameters, there is no paradox, right? It is simply unclear as to which question the OP is asking (although I would argue that the question as posed, "What is the probability that the coin came up heads", infers that they mean the absolute state of the coin, regardless of which caused her to wake, such that the answer is 1/2 - but okay, semantics). It appears that the only problem is in the clarity of the question stated by the OP, not what is actually going on mathematically, correct?

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Agreed. In my third reply in this thread, after I had read the link PTB gave in which an argument for the answer being 1/3 was given, I wrote:

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I think the problem, as stated, was ambiguous -- or rather, not as ambiguous as is required for the "paradox". This part here:

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On Sunday evening we ask Beauty what she thinks the probability is of the coin landing heads. Unsurprisingly she says 1/2. Whenever she wakes up she will say 1/3.

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The probability of the coin landing heads??? 1/2. The probability of it having landed heads?? 1/3 is a valid answer.

The reason 1/3 is now valid is because it can be implied that the question is actually: "what is the probability that the coin landed heads and then we woke you and asked you this question?"

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In that article, the question to Beauty is asked as: "When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?"

The key word is "is". It's ambiguous as to what probability space Beauty should be using. The coin-flip space, or the coin-flip-awaking space.

So, I think we all agree. No paradox, for sure. [img]/images/graemlins/smile.gif[/img]

[f97tosc]

One more explanation here, in favor of 1/3.

First let's disregard the coin, and simply wake her up Monday and Tuesday, and she loses memory in between. And she knows this. Then when she wakes up, she will assign the probability 0.5 to it being Monday or Tuesday.

Suppose now that we independently of this throw the fair coin once (she will still wake up both days regardless of outcome). Now clearly when she wakes up she will assign the probability 0.25 to each of the 4 combinations in (M,T)x(H,T).

Now suppose further that after we wake her up, for the specific case Tuesday and Heads, we tell her, "go back to sleep" but for any other combination we ask her "what is the probability for heads?", and that she knows this.

Hopefully, we can all agree that for this version of the problem, she will arrive at 1/3, by Bayes' theorem, when she gets the question.

And the only difference between this problem, and the stated problem, is that in the stated problem, she never gets to wake up on Tuesday. But I don't see how that should change her information when she is woken up. It may throw us off a bit since it distorts the meaning of an 'observation' (did she get to 'observe' what happened the second day?).

Intuitively speaking, when woken up, she learns that she was not told "go back to sleep" in the modified problem, or that she "is not asleep" in the original problem.

[KipBond]

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Beauty's awakenings are exactly the same as the balls in the bag.

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I disagree. Beauty is being woken twice and asked P(H) after a tails event, thus giving her more information about the P(H). In your ball experiment, the question was only asked once for both outcomes; no new information.

If I only asked you the P(H) if the result was tails, it's easy to see how my asking you the P(H) is in fact more information to determine P(H). (Assuming the event space is understood to be P(H & I'm asking you this question)).

[jason1990]

I thought I would be away, and yet I am still here. The internet is like the moon. It follows you wherever you go. Anyway, my ability to read and post here may be a bit sporadic in the next couple of weeks as I am traveling. So if my replies are delayed or nonexistent, please do not be offended.

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jason1990,

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P(H|W) = P(H)P(W|H)/[P(H)P(W|H) + P(T)P(W|T)]

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Is that really derived from Bayes Theorem? The answer on the right side is "1 + something"!

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Actually, this is Bayes theorem itself, at least the form that I typically see presented in the textbooks I teach from. The right side is of the form a/(a + b). It is a true statement.

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I agree with your conclusion about the paradox though - ish. I think 1/2 is her credence and 1/3 her betting odds

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Frankly, I do not think she has well-defined "betting odds" here. Her betting odds would depend on how often she is forced to bet. The experimenters can freely manipulate that and she will never know because of the amnesia. A person's betting odds are supposed to be tuned to the precise amount that protects the person from being exploited. There is no way to protect herself from exploitation here, no matter what odds she chooses. I think betting odds are not the right concept for this problem.

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Once the assumptions are in place and the experiment begins, the Bayesian is bound by the formal rules of probability. And the formal rules of probability say that you condition on events or information (sigma-algebras). You do not take "no information", convert it into assumptions, and then condition on that. The place for converting "no information" into assumptions is in the beginning, when you choose your prior and build your probability space.

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Beauty isn't making assumptions.

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My reply was hasty and my language informal. I chose my language because I thought PairTheBoard would understand it, based on our previous conversations. Here is a less informal, but still non-rigorous, explanation of what I meant. When I said Elga was making assumptions, I meant he was defining primitive credences. SB's belief set is nothing more than an assignment of probabilities to the events in some probability space. If we assign a few of these (which I am informally calling the primitive ones), then the rest will be forced on us by the laws of probability and the requirement that SB's beliefs are consistent. To show that 1/2 before the experiment and 1/3 during are consistent, we must build a probability space that encodes both of these probabilities. Elga does not do that. Elga (implicitly) builds a probability space that encodes only the 1/3 probability. His sample space (apparently) is {H1,T1,T2}. He then gives an argument for why it is reasonable to assign 1/3 to each outcome in this space. But he cannot formulate the 1/2 before probability in this space. So his argument is incomplete. It does not show that these two credences are consistent. To show they are consistent, we must build a probability space that models the entire experiment from beginning to end. We assign the primitive credences when we build that model. Elga builds a probability space that only models the middle of the experiment.

[jason1990]

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Suppose we do the following experiment instead. First we flip the coin. Then if it is T, we put two black tokens in an urn. If its [H], we put one black and one white token in the urn.

Now, let S take one token and guess if it is H or T. Obviously, [if] it is black, she should give the probability one third to H.

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I do not believe that this is a fair translation of the problem. In your urn model, seeing a black token would be genuinely new information, precisely because of the fact that she knew it was possible to see a white token.

I think a fairer translation would be this. Flip the coin and fill your urn exactly as you describe. But instead of letting S take a token, the experimenter looks inside the bag, takes out a black token, and shows it to her. In that case, the answer is still 1/2.