🏅 Top traders

What "Change my mind" tag really means : "My mind can't be changed".

@Zardoru stay salty 😜

Pretty sure thirder is correct for the most intuitive/relevant meanings of probability, but just have two minutes before this market closes so don't have time to write up my understanding. Here's a copy-paste of an interesting framework from a paper posted below:

For this purpose, imagine a “companion” of Beauty, the Prince, who is also put to sleep on Sunday, and woken on Monday. However, unlike Beauty, the Prince is woken on Tuesday regardless of how the coin lands, after being administered a drug that causes him to forget his Monday awakening. Like Beauty, he is always woken on Wednesday and told the experiment is over. Beauty and the 15 Prince will be in the same room at all times, and will be free to discuss their situation (if both are awake). All this is known to both Beauty and the Prince. If the Prince is woken before Wednesday and finds that Beauty has also been woken, what probability should he assign to the coin landing Heads? Quite clearly, he should assign probability 1/3 to Heads, since given Heads, the probability of Beauty being woken with him is 1/2, whereas given Tails this probability is 1. Since the coin is fair, this factor of two larger probability of what is observed given Tails should produce a factor of two larger probability of Tails given the observation that Beauty has been woken too. The Prince will tell Beauty of his conclusion. Should Beauty disagree, and maintain that the probability of Heads is actually 1/2? Beauty and the Prince have the same information (and even if they didn’t, they would after discussing the situation). Beauty will agree that the Prince’s reasoning is correct, for him. There seem to be no grounds for her to decide that, for her, the probability should be different. Furthermore, if we wish, we can disallow discussions between Beauty and the Prince — Beauty is intelligent enough to know what the Prince’s conclusion will be without him having to tell her. Indeed, we can assume that the Prince is hidden from Beauty by a curtain, as long as she knows he is there. Does it really matter if we go one step further and eliminate the Prince altogether?

Market closes in 9 hours guys, last chance to add something.

The thirder argument on wikipedia seems to be based on the fact that:

P(Tails | Monday) = P(Heads | Monday)

What is the best argument that this is correct?

@levifinkelstein here are three nice thirder arguments, quoted from the Winkler paper I cited (p. 580). The Symmetry argument sounds like the one you're referring to.

Repetition. Repeat the experiment 100 times; then SB will be awakened about 150 times, 50 of them to Heads. So SB’s probability of Heads, on awakening, must be 1/3.

Gambling. Ask SB, upon each awakening, if she’s willing to have $3 deducted from her bank account if the coin landed Heads, provided that $2 is added to her account if the coin landed Tails. (She understands that if it’s Tuesday morning and she accepted the bet on Monday, her holdings may already have changed—but that should have no influence on today’s decision.) As a thirder, SB should accept the bet: Her expectation is 1/3(−$3) + 2/3(+$2) > 0. And she’s right to do so: Over the course of the experiment, she ends up $4 ahead if the coin landed Tails and only $3 behind if it landed Heads. A corresponding calculation by the halfer SB would lead her to refuse the bet.

Symmetry. Suppose that 15 minutes after her Monday awakening, SB will be told what day it is. If she hears it’s Monday, her probability of Heads is 1/2. Suppose instead SB will be told the result of the coin flip. Then, if it’s Tails, her probability that it’s Monday is 1/2. The first implies P(Heads and Monday) = P(Tails and Monday); the second, that P(Tails and Monday) = P(Tails and Tuesday). These three probabilities represent exhaustive and mutually exclusive events, thus each is equal to 1/3.

@WilliamDAlessandro For the symmetry one, why is it the case that P(Heads | Monday) = 1/2?

@levifinkelstein because presumably P(Heads | Monday) should equal P(Heads). Learning that it's Monday doesn't give SB any additional information about the result of the flip, since she'll be woken on Monday regardless of the result.

@WilliamDAlessandro "P(Heads | Monday) should equal P(Heads)" I don't see why this is true. The fact that you've sampled an awakening and it's not Tuesday seems like it should count as evidence for it being Heads since Tuesday is less likely in the universe where the coin landed heads.

@levifinkelstein It's like if you have 2 urns, one has balls labeled A and B and the other has a single ball labeled A. If you don't know which urn is which and you draw a ball from one of them and it's an A then this should update you towards that it was drawn from the 1 ball urn.

@levifinkelstein I think I've got it! I stayed up late last night trying to work out your urn example. For a while I was confused because it felt like urns should be the "day" equivalent and balls should be the "coin" equivalent but the way you've defined it it's the other way around, but that's all fine.

The key point is that yes, sampling an awakening and finding it to be Monday does increase your credence that it is Heads. It increases your credence from 1/3 to 0.5.

I think the way @WilliamDAlessandro stated the symmetry argument might not have been quite right, or at least the explanation that "P(Heads | Monday) should equal P(Heads)" isn't quite right. (Should that "P(Heads)" be P(Heads | Taking part in SB experiment) or P(Heads | Single fair coin flip)? The whole point of the Thirder position is that those aren't the same!)

Now, I've not proved the symmetry argument there - I've just salvaged it from being self-inconsistent (the worry that was keeping me up last night). At the end of the day it still rests on the premise that P(Heads | Monday) = P(Tails | Monday) (and therefore both are equal to 0.5). The Halfer argument must explicitly reject this premise. The Bayesian argument (that sampling that it's Monday increases your credence of Heads) doesn't get you there on its own because both Halfer and Thirder arguments are consistent with it. (They just have different priors.)

Your forensic questioning seeded enough doubt in my mind to make me sell my Thirder shares. I still believe Thirder to be correct, but it's not as clear-cut as I previously thought.

@Fion "(They just have different priors.)" what are the different priors in this case?

@levifinkelstein Priors in the Bayesian sense. In terms of SB it's the final answer. Thirders say that the prior P(H | taking part in SB) = 1/3. Halfers say it's 1/2.

Then I still don't understand "At the end of the day it still rests on the premise that P(Heads | Monday) = P(Tails | Monday)" how this premise is justified. Where does it come from? The symmetry thing doesn't seem to make sense.

@levifinkelstein The symmetry argument is the most complicated of the three William quoted. If you're not sure about it I'd recommend the other two. Personally I find "P(Heads | Monday) = P(Tails | Monday)" to be intuitive and more natural than the alternative, but I don't think it can be proven in an obviously satisfying way. At some point you need to make an assumption about what conditional probability means.

(Necessary disclaimer that I'm not a mathematician.)

@Fion The repetition thing doesn't make sense to me. Imagine you flip a coin, if it lands heads you eat an apple, if it lands tails you eat two apples. If you run the experiment 100 times you will have eaten about 150 apples, 50 of them from heads. This means the probability of heads is 1/3! But does this make sense? Aren't you just double counting one event?

"The usual operational definition of credence is that it's positive EV to take cash bets if and only if offered odds better than the credence, assuming you know the cash bet would be offered in all situations that match your current observations."

We need a clarification on what Levi is taking credence to be because this is not at all what I would think of it as.
If you think of the problem of "assuming you could make $1 on each guess, what should you guess?" then obviously the answer is tails since you will get twice as many guesses if it hits tails. If I ask you to guess if a coin comes up heads or tails and if you're right about tails I'll give you $100 and if you're right about heads I'll give you $1 then your EV is tails favored but your credence is still 50/50.

@Zardoru makes a similar argument to the betting one as far as I can tell. The other arguments are in the form of papers which I cba reading, if there is a concise explanation for the thirder position then posting it would be much appreciated.

My argument for the halfer position is: The coin is fair, sleeping beauty knows the coin is fair and gets no new information.

The answer to that when seeing other people discussing this is to say that sleeping beauty gets some information from some anthropological principle thing by waking up. I don't understand the argument for why people think this makes sense, it's pretty fuzzy to me so perhaps someone is able to post an argument making the case.

Here is the argument for the thirder position from wikipedia:

"The thirder position argues that the probability of heads is 1/3. Adam Elga argued for this position originally[2] as follows: Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By even a highly restricted principle of indifference, given that the coin lands tails, her credence that it is Monday should equal her credence that it is Tuesday, since being in one situation would be subjectively indistinguishable from the other. In other words, P(Monday | Tails) = P(Tuesday | Tails), and thus

P(Tails and Tuesday) = P(Tails and Monday).

Suppose now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. Guided by the objective chance of heads landing being equal to the chance of tails landing, it should hold that P(Tails | Monday) = P(Heads | Monday), and thus

P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday)."

The problem is claiming P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday)
Here I'm going to assume that P(Tails) = P(Heads) = 50% and it isn't saying P(Heads) = P(Heads | Waking up) since that'd beg the question.

What is P(Monday)? Seems like it would be 75%. If it was heads it is 100% monday, if it was tails it's 50% monday.
What is P(Tails | Monday)? 50% since it was either heads or tails
What is P(Tails | Tuesday)? 100% since it must be tails
What is P(Heads | Tuesday)? 0%

What is P(Heads | Monday)? 50%
What is P(Monday | Tails)? which should be 50%

P(Tails and Monday) = P(Monday | Tails) * P(Tails) = 50% * 50%= 25%
P(Tails and Tuesday) = P(Tuesday | Tails) * P(Tails) = 50% * 50% = 25%

P(Heads and Monday) = P(Monday | Heads) * P(Heads) = 100% * 50% = 50%

P(Heads and Tuesday) = P(Tuesday | Heads) * P(Heads) = 0% * 50% = 0%

When wikipedia says P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday) This is wrong.

Would appreciate thirders weighing in on me making some error or bad assumption.

https://arxiv.org/abs/math/0608592
I show that these problems are satisfyingly resolved by applying the principle that one should always condition on all evidence - not just on the fact that you are an intelligent observer, or that you are human, but on the fact that you are a human with a specific set of memories. This Full Non-indexical Conditioning' (FNC) approach usually produces the same results as assuming both SSA and SIA, with a sufficiently broad reference class, while avoiding their ad hoc aspects. I argue that the results of FNC are correct using the device of hypothetical `companion'' observers, whose existence clarifies what principles of reasoning are valid.

@JacobPfau I will not read this, you'll have to explain the argument here.

This 2017 paper by Peter Winkler (a mathematician) is a really nice opinionated survey with a good discussion of how thirders might answer the main halfer argument. As Winkler says:

"[T]o the extent that there is agreement about what is asked, [Sleeping Beauty is] a mathematical question to which many think the straightforward answer is 1/3. Some consequences of the 1/3 answer appear surprising at first, but upon scrutiny, seem (for some) increasingly intuitive. In particular, being conscious at a given moment may constitute legitimate information, even if—and in some cases, especially if—the moment’s time label is not known."

@WilliamDAlessandro I will not read the paper, you'll have to explain it here.

The usual operational definition of credence is that it's positive EV to take cash bets if and only if offered odds better than the credence, assuming you know the cash bet would be offered in all situations that match your current observations.

The nice thing about the operational view of credence is you can just work out the math. Imagine if the interviewer promised that every time he wakes you up, you will be offered a bet where you put up $2 against the experimenter's $1.01, where you get paid out if the coin came up tails and the experimenter gets paid out if it came up heads. The strategy "always accept that bet" is positive EV -- in universe A where the coin came up heads, you lose $2, in universe B where it came up tails, you gain $1.01*2 = $2.02, because you're asked to bet twice.

So the operational view of credence says you should be a thirder.

(The reason people argue about the sleeping beauty problem is that some people's intuition for credence doesn't match the operational definition. Some people's intuition for credence is something like "in how many universes are you right". If you imagine a payoff graph that matches that -- e.g. where the interviewer put $1.01 into a box once at the start of the experiment, and offers you a bet where you put up $2 vs. taking whatever's currently in the box -- you'll get a higher payoff using 1/2 than 1/3 to decide whether to take that bet, because the second time you're offered the bet in universe B the box is empty.

In some sense this comes down to what you're trying to optimize. If you're trying to answer a question where it matters whether you get it right both times you wake up, you should use the 1/3 number. If you're trying to answer a question where it only matters once per universe whether you were right, you should use the 1/2 number. But I think the 1/3 view is closer to the normal definition of credence where you're using bets that matter equally in all situations that match current observations.)

@MichaelLucy I don't quite see how this argues for the thirder position, given that it's a claim about probability.

If I have a fair coin that costs $1 to flip and I get $10 if it lands tails. How does the fact that I'm expected to earn money taking the bet say anything about the probability of tails?

@levifinkelstein I realize the market is closed, but if you're still interested:

The point of a probability is you can multiply it by the payout to see which side to take. So you're trying to map from a (probability, payout) pair to a decision.

In the case of the fair coin, you want to take whichever payout is higher, which is consistent with assigning 0.5 to each side of the coin -- if offered a payout graph of (0.5*9, 0.5*-1), it's clear you should take it. 0.5 is the correct probability because it multiplies out correctly for cases like $1 to play and $2.01 for tails.

In the sleeping beauty problem, the question is "If when you wake up you observe being offered a bet that costs $1 and pays out $1.9 if the sleeping coin came up tails, should you take it?"

If the answer is yes, I think you're a thirder, i the answer is no I think you're a halfer.

The thing that makes it complicated is that all the reasoning about probabilities and payouts only works if you don't think the person offering the bet is deciding whether or not to offer it to you based on the outcome. E.g. if someone flips a coin, peaks at it, and then offers you a bet which is naively in-the-money, you can't reason about whether or not to take that bet without knowing whether they decided to offer you the bet based on what they saw.

The way around this is to just say that the person offering the bet isn't allowed to use any information they may know about which way the coin came up when deciding whether or not to offer the bet. If you have that assumption, you can do the naive probability * payout math and everything will be consistent with the real probability.

In the case of the sleeping beauty problem, that means the researcher can't make their decision about whether to offer you the bet based on their observation of the coinflip. In particular, in the tails case, they can't offer you the bet the first time you wake up and not the second, because when they choose not to offer it to you the second time, that case only exists in the tails universe, so they're implicitly deciding not to offer you the bet based on the coin's outcome. There's a causal arrow from "was the coin tails" to their branch on which time they're waking you up, so that branch isn't allowed.

Anyway, if you buy all of that, then I think the answer to the real question of "If when you wake up you observe being offered a bet that costs $1 and pays out $1.9 if the sleeping coin came up tails, should you take it, assuming the person offering the bet isn't conditioning on their knowledge of the sleeping coin result?" is "yes, obviously", you just work out the payoff, which means you should be a thirder.