I don’t understand what return ['Tails&Monday','Tails&Tuesday'] and ListC += outcome mean. Can you explain it more? Perhaps operationalize it into some specific way Sleeping Beauty should act in some situation?
For example, if Sleeping Beauty is allowed to make bets every time she is woken up, I claim she should bet as though she believes the coin came up with probability 1⁄3 for Heads and 2⁄3 for Tails (≈because over many iterations she’ll find herself betting in situations where Tails is true twice as often as in situations where Heads is true).
I don’t understand what your solution means for Sleeping Beauty. The best operationalization I can think of for “Ω={Heads&Monday, Tails&Monday&Tuesday}” is something like:
“Thanks for participating in the experiment Ms. Beauty, and just FYI it’s Tuesday”
Whether Beauty should bet each time she wakes up depends very critically on the rules of the wager. Some examples:
Rule 1: Independent awakening bets: Each awakening beauty can bet $1 on the outcome of the coin. The bets at each awakening all stand independently. -In this case she should bet as if it was a 2/3rd chance of tails. After 100 coin tosses she has awoken 150 times, and for 100 of them it was tails.
Rule 2: Last bet stands: Each awakening beauty can bet $1 on the outcome of the coin. Only beauty’s final decision for each toss is taken into account, for example any bet she makes Tuesday replaces anything decided on Monday. -She treats it as 50⁄50.
Rule 3: Guess Wrong you Die (GRYD): On each awakening beauty must guess the coins outcome. If she has made any incorrect guesses by the end of the run, she is killed. -She should pick either heads or tails beforehand and always pick that. Picking heads is just as good as picking tails.
The above set gives 1 “thirder’s game”, two “halfer’s games” and one that I can’t classify (GRYL). She will certainly find herself betting in twice as many tails situations as heads ones (hence the Rule 1 solution), but whether that should determine her betting strategy depends on the rules of the bet. As Ape in Coat has said Rule 1 can be interpreted as “50/50 coin, but your deposit and winnings are both doubled on tails” (because on tails Beauty makes two wagers).
It seems to me that Rule 1 is a direct translation of the Sleeping Beauty problem into a betting strategy question, while the other rules correspond to different questions where a single outcome depends on some function of the two guesses in the case of tails. Doing the experiment 100 times under that rule, Beauty will have around 150 identical awakening experiences. The payout for each correct guess is the same, $1, and the correct guess would be tails 2⁄3 of the time. So surely the probability that the coin had landed tails prior to these events is 2/3? Not because it’s an unfair coin or there was an information update (neither is true), but because the SB problem asks the probability from the perspective of someone being awakened, and 2⁄3 of these experiences happen after flipping tails. It seems a stretch to say the bet is 50⁄50 but the 2nd 50% happens twice as often.
There is no such thing as direct translation of a problem into a betting strategy question. A model for a problem should be able to deal with any betting schemes, no matter how extravagant.
And the scheme where the Beauty can bet on every awakening is quite extravagant. It’s an asymetric bet on a coin toss, where Tails outcome is rewarded twice as Heads outcome.
So surely the probability that the coin had landed tails prior to these events is 2/3? Not because it’s an unfair coin or there was an information update (neither is true)
If there is no information update then the probability of the coin to be Tails can’t change from 1⁄2 to 2⁄3. It would contradict the law of conservation of expected evidence.
but because the SB problem asks the probability from the perspective of someone being awakened, and 2⁄3 of these experiences happen after flipping tails.
As I’ve written in the Effects of Amnesia section, from Beauty’s perspective Tails&Monday and Tails&Tuesday awakening are still part of the same elementary outcome because she remembers the setting of the experiment. If she didn’t know that Tails&Monday and Tails&Tuesday necessary follow each other, if all she knew is that there are three states in which she can awaken, then yes, she should’ve reasoned that P(Tails)=2/3.
Alternatively if the question was about a random awakening of the Beauty among multiple possible experiments, then, once again, P(Heads) would be 1⁄3. But in the experiment as stated, the Beauty isn’t experiencing a random awakening, she is experiencing ordered awakening, determined by a coin toss.
I don’t understand what return ['Tails&Monday','Tails&Tuesday'] and ListC += outcome mean. Can you explain it more?
ListC += outcome is a list concatenation operation. Meaning, that both 'Tails&Monday' and 'Tails&Tuesday' are added to the list in one tick of a logical clock, thus preserving the order between them.
For example, if Sleeping Beauty is allowed to make bets every time she is woken up, I claim she should bet as though she believes the coin came up with probability 1⁄3 for Heads and 2⁄3 for Tails (≈because over many iterations she’ll find herself betting in situations where Tails is true twice as often as in situations where Heads is true).
I’ll be exploring betting odds in the next post in more details. But the short version is that she should bet according to 1⁄2 probability for Heads which naturally leads to 1:2 betting odds in the betting scheme that you’ve proposed. Probability for Heads and Tails are equal but successfully guessing Tails is rewarded twice as much as successfully guessing Heads.
I don’t understand what your solution means for Sleeping Beauty.
It means that during the experiment she can’t lawfully reason about such events as “This day is Monday” or “This day is Tuesday”, only “Monday awakening happens in this experiment” and “Tuesday awakening happens in this experiment”. Whether she will be able to reason about “this day” after the experiment is over is a separate question and depends on whichever circumstances she will be in.
During the experiment the Beauty tries to distinguish between two outcomes: “Heads, only Monday awakening” and “Tails, Monday and Tuesday awakenings” if she manages to find some evidence that Tuesday awakening is to happen in this iteration of the experiment she lawfully updates in favor of the second outcome. It doesn’t matter whether the evidence is about today or tomorrow.
For example, imagine that the Beauty awakens and finds in her room a timetable, that some of the scientists has left there by mistake. According to the timetable the Beauty is currently experiencing a Monday awakening but also has to be awakened tomorrow. The Beauty shouldn’t think like: “Oh, well, seems that today is Monday so both Heads and Tails are equally likely”. No! She should immediately conclude that the coin is Tails, even though she is not experiencing Tuesday awakening at this moment of physical time. Knowledge that it definitely happens in the future is enough.
You don’t have to reply, but FYI I don’t understand what ListC represents (a total ordering of events defined by a logical clock? A logical clock ordering Beauty’s thoughts, or a logical clock ordering what causally can affect what, or logically affect what allowing for Newcomb-like situations? Why is there a clock at all?), how ListC is used, what concatenating multiple entries to ListC means in terms of beliefs, etc. If it’s important for readers to understand this you might have to step us through (or point us to an earlier article where you stepped us through).
ListC represents the same thing as ListE, ListL and ListU. Its a list of ordered outcomes of multiple runs of a particular model, which than we can compare to ListSB—a list of ordered awakenings of the Beauty in multiple iterations of an experiment and see whether their statistical properties are the same. The nature of the test is described in Statistical Analysis part of the post. I provide an inner hyperlink to it here:
The correct model represents the only lawful way to reason about the Sleeping Beauty problem without smuggling unjustified assumptions that the current awakening is randomly sampled. It passes the statistical test with flying colors, as it essentially reimplements the sleepingBeauty() function
Right, I read all that. I still don’t understand what it means to append two things to the list.
Here’s how I understand modelLewis, modelElga, etc.
“This model represent the world as a probability distribution. To get a more concrete sense of the world model, here’s a function which generates a sample from that probability distribution”
Here’s how I understand your model.
“This model represents the world as a ????, which like a probability distribution but different. To get a concrete sense of the world model, here’s a function which generates a sample from that probability distribution JUST KIDDING here’s TWO samples”.
Why can you generate two samples at once? What does that even mean?? The world model isn’t quite just a stationary probability distribution, fine, what is it then? Your model isn’t structured like other models, fine, but how is it structured? I’m drowning in type errors.
EDIT and I’m suggesting be really concrete, if you can, if that will help. Like come up with some concrete situation where Beauty makes a bet, or says a thing, (“Beauty woke up on Monday and said ‘I think there’s a 50% chance the coin came up on heads, and refuse to say there’s a state of affairs about what day it presently is’”) and explain what in her model made her make that bet or say that thing. Or maybe draw a picture which what her brain looks like under that circumstance compared to other circumstances.
Here’s how I think of what the list is. Sleeping Beauty writes a diary entry each day she wakes up. (“Nice weather today. I wonder how the coin landed.”). She would like to add today’s date, but can’t due to amnesia. After the experiment ends, she goes back to annotate each diary entry with what day it was written, and also the coin flip result, which she also now knows.
The experiment is lots of fun, so she signs up for it many times. The Python list corresponds to the dates she wrote in her dairy.
I don’t understand what
return ['Tails&Monday','Tails&Tuesday']andListC += outcomemean. Can you explain it more? Perhaps operationalize it into some specific way Sleeping Beauty should act in some situation?For example, if Sleeping Beauty is allowed to make bets every time she is woken up, I claim she should bet as though she believes the coin came up with probability 1⁄3 for Heads and 2⁄3 for Tails (≈because over many iterations she’ll find herself betting in situations where Tails is true twice as often as in situations where Heads is true).
I don’t understand what your solution means for Sleeping Beauty. The best operationalization I can think of for “Ω={Heads&Monday, Tails&Monday&Tuesday}” is something like:
Whether Beauty should bet each time she wakes up depends very critically on the rules of the wager. Some examples:
Rule 1: Independent awakening bets: Each awakening beauty can bet $1 on the outcome of the coin. The bets at each awakening all stand independently. -In this case she should bet as if it was a 2/3rd chance of tails. After 100 coin tosses she has awoken 150 times, and for 100 of them it was tails.
Rule 2: Last bet stands: Each awakening beauty can bet $1 on the outcome of the coin. Only beauty’s final decision for each toss is taken into account, for example any bet she makes Tuesday replaces anything decided on Monday. -She treats it as 50⁄50.
Rule 2: Guess Right You Live (GRYL): On each awakening beauty must guess the coins outcome. If she has made no correct guess by the end of the run, she is killed. -For fair coin pick randomly between heads and tails, but for an unfair coin its a bit weird: https://www.lesswrong.com/posts/HQFpRWGbJxjHvTjnw/?commentId=BrvGnFvpK3fpndXGB
Rule 3: Guess Wrong you Die (GRYD): On each awakening beauty must guess the coins outcome. If she has made any incorrect guesses by the end of the run, she is killed. -She should pick either heads or tails beforehand and always pick that. Picking heads is just as good as picking tails.
The above set gives 1 “thirder’s game”, two “halfer’s games” and one that I can’t classify (GRYL). She will certainly find herself betting in twice as many tails situations as heads ones (hence the Rule 1 solution), but whether that should determine her betting strategy depends on the rules of the bet. As Ape in Coat has said Rule 1 can be interpreted as “50/50 coin, but your deposit and winnings are both doubled on tails” (because on tails Beauty makes two wagers).
It seems to me that Rule 1 is a direct translation of the Sleeping Beauty problem into a betting strategy question, while the other rules correspond to different questions where a single outcome depends on some function of the two guesses in the case of tails. Doing the experiment 100 times under that rule, Beauty will have around 150 identical awakening experiences. The payout for each correct guess is the same, $1, and the correct guess would be tails 2⁄3 of the time. So surely the probability that the coin had landed tails prior to these events is 2/3? Not because it’s an unfair coin or there was an information update (neither is true), but because the SB problem asks the probability from the perspective of someone being awakened, and 2⁄3 of these experiences happen after flipping tails. It seems a stretch to say the bet is 50⁄50 but the 2nd 50% happens twice as often.
There is no such thing as direct translation of a problem into a betting strategy question. A model for a problem should be able to deal with any betting schemes, no matter how extravagant.
And the scheme where the Beauty can bet on every awakening is quite extravagant. It’s an asymetric bet on a coin toss, where Tails outcome is rewarded twice as Heads outcome.
If there is no information update then the probability of the coin to be Tails can’t change from 1⁄2 to 2⁄3. It would contradict the law of conservation of expected evidence.
As I’ve written in the Effects of Amnesia section, from Beauty’s perspective Tails&Monday and Tails&Tuesday awakening are still part of the same elementary outcome because she remembers the setting of the experiment. If she didn’t know that Tails&Monday and Tails&Tuesday necessary follow each other, if all she knew is that there are three states in which she can awaken, then yes, she should’ve reasoned that P(Tails)=2/3.
Alternatively if the question was about a random awakening of the Beauty among multiple possible experiments, then, once again, P(Heads) would be 1⁄3. But in the experiment as stated, the Beauty isn’t experiencing a random awakening, she is experiencing ordered awakening, determined by a coin toss.
ListC += outcomeis a list concatenation operation. Meaning, that both'Tails&Monday'and'Tails&Tuesday'are added to the list in one tick of a logical clock, thus preserving the order between them.I’ll be exploring betting odds in the next post in more details. But the short version is that she should bet according to 1⁄2 probability for Heads which naturally leads to 1:2 betting odds in the betting scheme that you’ve proposed. Probability for Heads and Tails are equal but successfully guessing Tails is rewarded twice as much as successfully guessing Heads.
It means that during the experiment she can’t lawfully reason about such events as “This day is Monday” or “This day is Tuesday”, only “Monday awakening happens in this experiment” and “Tuesday awakening happens in this experiment”. Whether she will be able to reason about “this day” after the experiment is over is a separate question and depends on whichever circumstances she will be in.
During the experiment the Beauty tries to distinguish between two outcomes: “Heads, only Monday awakening” and “Tails, Monday and Tuesday awakenings” if she manages to find some evidence that Tuesday awakening is to happen in this iteration of the experiment she lawfully updates in favor of the second outcome. It doesn’t matter whether the evidence is about today or tomorrow.
For example, imagine that the Beauty awakens and finds in her room a timetable, that some of the scientists has left there by mistake. According to the timetable the Beauty is currently experiencing a Monday awakening but also has to be awakened tomorrow. The Beauty shouldn’t think like: “Oh, well, seems that today is Monday so both Heads and Tails are equally likely”. No! She should immediately conclude that the coin is Tails, even though she is not experiencing Tuesday awakening at this moment of physical time. Knowledge that it definitely happens in the future is enough.
You don’t have to reply, but FYI I don’t understand what
ListCrepresents (a total ordering of events defined by a logical clock? A logical clock ordering Beauty’s thoughts, or a logical clock ordering what causally can affect what, or logically affect what allowing for Newcomb-like situations? Why is there a clock at all?), howListCis used, what concatenating multiple entries toListCmeans in terms of beliefs, etc. If it’s important for readers to understand this you might have to step us through (or point us to an earlier article where you stepped us through).ListC represents the same thing as ListE, ListL and ListU. Its a list of ordered outcomes of multiple runs of a particular model, which than we can compare to ListSB—a list of ordered awakenings of the Beauty in multiple iterations of an experiment and see whether their statistical properties are the same. The nature of the test is described in Statistical Analysis part of the post. I provide an inner hyperlink to it here:
Right, I read all that. I still don’t understand what it means to append two things to the list.
Here’s how I understand
modelLewis,modelElga, etc.“This model represent the world as a probability distribution. To get a more concrete sense of the world model, here’s a function which generates a sample from that probability distribution”
Here’s how I understand your model.
“This model represents the world as a ????, which like a probability distribution but different. To get a concrete sense of the world model, here’s a function which generates a sample from that probability distribution JUST KIDDING here’s TWO samples”.
Why can you generate two samples at once? What does that even mean?? The world model isn’t quite just a stationary probability distribution, fine, what is it then? Your model isn’t structured like other models, fine, but how is it structured? I’m drowning in type errors.
EDIT and I’m suggesting be really concrete, if you can, if that will help. Like come up with some concrete situation where Beauty makes a bet, or says a thing, (“Beauty woke up on Monday and said ‘I think there’s a 50% chance the coin came up on heads, and refuse to say there’s a state of affairs about what day it presently is’”) and explain what in her model made her make that bet or say that thing. Or maybe draw a picture which what her brain looks like under that circumstance compared to other circumstances.
Here’s how I think of what the list is. Sleeping Beauty writes a diary entry each day she wakes up. (“Nice weather today. I wonder how the coin landed.”). She would like to add today’s date, but can’t due to amnesia. After the experiment ends, she goes back to annotate each diary entry with what day it was written, and also the coin flip result, which she also now knows.
The experiment is lots of fun, so she signs up for it many times. The Python list corresponds to the dates she wrote in her dairy.