You could have that conversation, but you don’t have to. The argument for assigning 50% to red is that it’s the only question Omega has asked you. There are several ways out of that. The first one is that the moment he offers you a 25% bet I would update to presume that 3:1 is not a positive e.v. bet, with a new number of perhaps 12.5% with a range of 0% to 25% with symmetric distribution. Similarly, if he offered me three to one that it wasn’t red, I would presume that it probably will be. On a similar note, when he asks about blue (even without any bets involved) I can’t see answering higher than 33.3%.
Contrast this with Alicorn watching this incident and offering me 3:1 after Omega asks my probability for red and I say 50%. I still have to update for Alicorn’s opinion, but I might or might not accept that bet.
The estimate should take into account the expectation of being asked further questions. The ignorance prior is applied to a model of observation. The model of observation expresses which questions you may be asked, and what structure will the dependencies between these possible observations have.
The estimate should take into account the expectation of being asked further questions.
I do not know how related this is to your comment, but it made me think of another response to the Dutch book objection. (Am I using that term correctly?)
If Omega asks me about a red bead I can say 100%. If he then asks about a blue bead I can adjust my original estimate so that both red and blue are the same at 50⁄50. Every question asked is adding more information. If Omega asks about green beads all three answers get shifted to 1⁄3.
This translates into an example with numbered balls just fine. The more colors or numbers Omega asks about decreases the expected probability that any particular one of them will come out of the jar simply because the known space of colors and numbers is growing. Until Omega acknowledges that there could be a bead of that color or number there is no particular reason to assume that such a bead exists.
If the example was rewritten to simply say any type of object could be in the jar, this still makes sense. If Omega asks about a red bead, we say 100%. If Omega asks about a blue chair, both become 50%. The restriction of colors and numbers is our assumed knowledge and has nothing to do with the problem at hand. We can meta-game all we want, but it has nothing to do with what could be in the jar.
The state of the initial problem is this:
A red bead could be in the jar
After the second question:
A red bead could be in the jar
A green bead could be in the jar
I suppose it makes some sense to include an “other” category, but there is no knowledge of anything other than red and green beads. The question of probability implies that another may exist, but is that enough to assign it a probability?
Every question asked is adding more information. If Omega asks about green beads all three answers get shifted to 1⁄3.
I don’t think we should treat omega as adding (much) new information with each question.
Omega is super intelligent, we should assume that he’s already went all the way down the rabbit hole of possible colors, including ones that our brains could process but our eyes don’t see. We’re not inferring anything about his state of mind because he’s only asking questions about red, green, and blue. A sequence of lilac turquoise turquoise lilac lilac says very much more about what’s in the jar than the two hundred color questions omega asked you beforehand.
Not every question Omega could ask would provide new information, but some certainly do. Suppose his follow-up questions were “What is the probability that the bead is transparent?”, “What is the probability that the bead is made of wood?” and “What is the probability that the bead is striped?”. It is very likely that your original probability distribution over colors implicitly set at least one of these answers to zero, but the fact that Omega has mentioned it as a possibility makes it considerably more likely.
If Omega asking if the bead could be striped changes you probability estimates, then you were either wrong before or wrong after (or likely both)
If omega tells you at the outset that the beads are all solid colors, then you should maintain your zero estimate that any are striped. If not, then you never should have had a zero estimate. He’s not giving you new information, he’s highlighting information you already had (or didn’t have.)
I don’t see any way to establish a reliable (non-anthropomorphic) chain of causality that connects there being red beads in the jars with Omega asking about red beads. He can ask about beads that aren’t there, and that couldn’t be there given the information he’s given you.
When Omega offered to save x+1 billion people if the earth was less than 1 million years old, I don’t think anyone argued that his suggesting it should change our estimates.
I don’t see any way to establish a reliable (non-anthropomorphic) chain of causality that connects there being red beads in the jars with Omega asking about red beads.
If I initially divide the state space into solid colours, and then Omega asks if the bead could be striped, then I would say that’s a form of new information—specifically, information that my initial assumption about the nature of the state space was wrong. (It’s not information I can update on; I have to retrospectively change my priors.)
Of note, I was operating under a bad assumption with regards to the original example. I assumed that the set was a finite but unknown set of colors or an infinite set of colors. In the former case, every question is giving a little information about the possible set. In the latter it really does not matter much.
A sequence of lilac turquoise turquoise lilac lilac says very much more about what’s in the jar than the two hundred color questions omega asked you beforehand.
Yes, this is true. Personally, I am still curious about what to do with the two hundred color questions.
Don’t think of probability as being mutable, as getting updated. Instead, consider a fixed comprehensive state space, that has a place on it for every possible future behavior, including the possible questions asked, possible pieces of evidence presented, possible actions you make. Assign a fixed probability measure to this state space.
Now, when you do observe something, this is information, an event, a subset on the global state space. This event selects an area on it, and encompasses some of the probability mass. The statements, or beliefs (such as “the ball #2 will be red”), that you update on this info, are probabilistic variables. A probabilistic variable is a function that maps the state space on a simpler domain, for example a binary discrete probabilistic variable is basically an event, a subset of the state space (that is, in some states, the ball #2 is indeed defined to be red, these states belong to the event of ball #2 being red).
Your info about the world retains only the part of the state space, and within that part of the state space, some portion of the probability mass goes to the event defining your statement, and some portion remains outside of it. The “updating” only happens when you focus on this info, as opposed to the whole state space.
If that picture is clear, you can try to step back to consider what kind of probability measure you’d assign to your state space, when its structure already encodes all possible future observations. If you are indifferent to a model, the assignment is going to be some kind of division into equal parts, according to the structure of state space.
IAWYC, but as pedagogy it’s about on the level of “How should you imagine a 7-dimensional torus? Just imagine an n-dimensional torus and let n go to 7.”
What if Omega wants you to commit to a bet based on your probabilities at every step?
Or what if he just straight up asks you what color you want to guess the bead will be, without asking about any individual colors? (Then you’d probably be best served by switching to a language with fewer basic color words, but that aside...)
What if Omega wants you to commit to a bet at every step?
Than you are forced to bid 0 because you have to account for any further questions, which sounds similar to what Vladimir_Nesov said.
By the way, I think adding another restriction to your example to force it back into your specific response is not particularly meaningful. In the case where you do not have to commit to a bet at every step, does what I say make sense? If so, than what Vladimir_Nesov suggested seems to be on the right path with regards to your restrictions.
Or what if he just straight up asks you what color you want to guess the bead will be, without asking about any individual colors? (Then you’d probably be best served by switching to a language with fewer basic color words, but that aside...)
Switching languages is a semantic trick. If we are allowed to use any words to describe the bead we can just say “not-clear” because the space of “not-clear” covers what we generally mean by “color”. We may as well say “the bead will be a colored bead.” All of this breaks the assumed principle of no information.
If Omega wanted a particular color and forced us into actually answering the annoying question, we are completely off the path of probabilities and it does not matter what you answer as long as you picked a color. If Omega then asked us what the probability of that particular color coming out of the jar would be, the answer should be the same as if you picked any other color. This drops to zero unless you self-restrict by the number of colors you can personally remember.
MrHen, whatever strategy you’re employing here, it doesn’t sound like a strategy for arriving at the really truly correct answer, but some sort of clever set of verbal responses with a different purpose entirely. In real life, just because Omega asked if the bead is red simply does not mean there is probability 0 of it being green.
MrHen, whatever strategy you’re employing here, it doesn’t sound like a strategy for arriving at the really truly correct answer, but some sort of clever set of verbal responses with a different purpose entirely.
Mmm… I was not trying to employ a strategy with clever verbal responses. I thought I was arguing against that, actually, so I must be far from where I think I am.
I feel like I am trying to answer a completely different question than the one originally asked. Is the question:
Knowing nothing about what is in the jar except that its contents are divided by color as per our definition of “color”, what is the probability of a red bead being pulled?
Knowing nothing about what is in the jar, what is the probability of a red bead being pulled?
I admittedly assumed the latter even though the article used words closer to the former. Perhaps this was my mistake?
In real life, just because Omega asked if the bead is red simply does not mean there is probability 0 of it being green.
I would agree. I do think that Omega asking about a red bead implies nothing about the probability of it being green. What I am currently wondering is if the question implies anything about the probability of the bead being red. If Omega acknowledges that the bead could be red, does that give red a higher probability than green?
I suppose I instinctively would answer affirmatively. The reasoning is that “red” is now included in the jar’s potential outcomes while green has not been acknowledged yet. In other words, green doesn’t even have a probability. Strictly speaking, this makes little sense, so I must be misstepping somewhere. My hunches are pointing toward my disallowing green into the potential outcomes.
This does not mean that I refuse to think of green as a color, but that green is not automatically included in the jar’s potential outcomes just because Omega used the word “color”. Is this the verbal cleverness you were referring to?
(Switching thoughts) In terms of arriving at the really truly correct answer, it seems that a strategy that gets closer as more beads is what is desired. If no beads are revealed, what sort of strategy is possible? I think the answer to this revolves around my potential confusion of the original question.
I apologize if I am mudding things up and am way off base.
Omega: So you would consider it more than fair if I offered you three dollars if the bead is red, and you paid me a dollar if it was non-red?
Me: No, because you have more information than I do, and the fact that you would offer this bet is evidence that I should use to update my epistemic probabilities.
Well, in the case of Omega, I would at least suspect that he intends to demonstrate that I am vulnerable to a Dutch book, even though he doesn’t need the money.
If you meet an Omega, that is pretty good evidence that you are living in a simulation: specifically, you are being simulated inside a philosopher’s brain as a thought experiment.
Sorry, you haven’t convincingly demonstrated the wrongness of 50%. MrHen’s position seems to me quite natural and defensible, provided he picks a consistent prior. For example, I’d talk with Omega exactly as you described up to this point:
...Omega: What is the probability of the first bead being blue as opposed to non-blue?
Me: 25%.
You ask why 25%? My left foot said so… or maybe because Omega mentioned red first and blue second. C’mon Dutch book me.
I think that doing it this way assumes that Omega is deliberately screwing with you and will ask about colors in a way that is somehow germane to the likelihood. Assume he picked “red” to ask about first at random out of whatever colors the beads come in.
This new information gives me grounds to revise my estimates as Omega asks further questions, but I still don’t see how it demonstrates the wrongness of initially answering 50%.
The reason 50⁄50 is bad is because the beads in the jar come in no more than 12 colors and we have no reason to favor red over the other 11 colors.
Knowing there is a cap of 12 possible options, it makes intuitive sense to start by giving each color equal weights until more information appears. (Namely, whenever Omega starts pulling beads.)
I suppose the relevant question is now, “Does Omega mentioning red tell us anything about what is in the jar?” When we know the set of possible objects in the jar, it really tells us nothing new. If the set of possible objects is unknown, now we know red is a possibility and we can adjust accordingly.
The assumption here is that Omega is just randomly asking about something from the possible set of objects. Essentially, since Omega is admitting that red could be in the jar, we know red could be in the jar. In the 12 color scenario, we already know this. I do not think that Omega mentioning red should effect our guess.
All this arguing about priors eerily resembles scholastics, balancing angels on the head of a pin. Okay I get it, we read Omega’s Bible differently: unlike me, you see no symbolic significance in the mention of red. Riiiiight. Now how about an experiment?
All this arguing about priors eerily resembles scholastics, balancing angels on the head of a pin. Okay I get it, we read Omega’s Bible differently: unlike me, you see no symbolic significance in the mention of red. Riiiiight. Now how about an experiment?
Agreed. For what it is worth, I do see some significance in the mention of red, but cannot figure out why and do not see the significance in the 12 color example. This keeps setting off a red flag in my head because it seems inconsistent. Any help in figuring out why would be nifty.
In terms of an experiment, I would not bet at all if given the option. If I had to choose, I would choose whichever option costs less and right it off as a forced expense.
In English: If Omega said he had a dollar claiming the next bead would be red and asked me what I bet I would bet nothing. If I had to pick a non-zero number I would pick the smallest available.
I think Alicorn is operating under a strict “12 colors of beads” idea based on what a color is or is not. As best as I can tell, the problem is essentially, “Given a finite set of bead colors in a jar, what is the probability of getting any particular color from a hidden mixture of beads?” The trickiness is that each color could have a different amount in the jar, not that there are any number of colors.
Alicorn answered elsewhere that when the jar has an infinite set of possible options the probability of any particular option would be infinitesimal.
If the number of possible outcomes is finite, fixed and known, but no other information is given, then there’s a unique correct prior: the maxentropy prior that gives equal weight to each possibility.
(Again, though, this is your prior before Omega says anything; you then have to update it as soon as ve speaks, given your prior on ver motivations in bringing up a particular color first. That part is trickier.)
(Again, though, this is your prior before Omega says anything; you then have to update it as soon as ve speaks, given your prior on ver motivations in bringing up a particular color first. That part is trickier.)
How would you update given the following scenarios (this is assuming finite, fixed, known possible outcomes)?
Omega asks you for the probability of a red bead being chosen from the jar
Omega asks you for the probability of “any particular object” being chosen
Omega asks you to name an object from the set and then asks you for the probability of that object being chosen
I don’t think #2 or #3 give me any new relevant information, so I wouldn’t update. (Omega could be “messing with me” by incorporating my sense of salience of certain colors into the game, but this suspicion would be information for my prior, and I don’t think I learn anything new by being asked #3.)
I would incrementally increase my probability of red in case #1, and decrease the others evenly, but I can’t satisfy myself with the justification for this at the moment. The space of all minds is vast; and while it would make sense for several instrumental reasons to question first about a more common color, we’re assuming that Omega doesn’t need or want anything from this encounter.
In the real-life cases which this is meant to model, though, like having a psychologist doing a study in place of Omega, I can model their mind by mine and realize that there are more studies in which I’d ask about a color I know is likely to come up, than studies in which I’d pick a specific less-likely color, and so I should update p(red) positively.
The fact that Omega is speaking English and uses the word “red” as opposed to “scarlet” or something is decent evidence that there are twelve colors in beadspace.
Right you are. I didn’t read the original problem carefully enough…
Nevertheless, you can replace “transparent” with a surprising color like lilac, fuchsia, or, um, cyan to restore the effect. The point is that even decent evidence that there are twelve colors in beadspace doesn’t justify a probability distribution on the number of colors that places all of its mass at twelve.
The twelve basic colors are so called because they are not kinds of other colors. Lilac and fuchsia are kinds of purple (I guess you could argue that fuchsia is a kind of red, instead, but pretend you couldn’t), and cyan is a kind of blue. Even if you pull out a navy bead and then a cyan bead, they are both kinds of blue in English; in Russian, they would be different colors as unalike as pink and red.
So you’re arguing that by definition, the basic color words define a mutually exclusive and exhaustive set. But there are colors near cyan which are not easy to categorize—the fairest description would be blue-green. In the least convenient world, when Omega asks you for odds on blue-green, you ask it if that color counts as blue and/or green, and it replies, “Neither; I treat blue-green as distinct from blue and green.” Then what do you do?
I was mentally categorizing that as “Omega deliberately screwing with you” by using English strangely, but perhaps that was unmotivated of me. But this gets into a grand metaphysical discussion about where colors begin and end, and whether there is real vagueness around their borders, and a whole messy philosophy of language hissy fit about universals and tropes and subjectivity and other things that make you sound awfully silly if you argue about them in public. I ignored it because the idea of the post wasn’t about colors, it was about probabilities.
That’s a shame, because uncertainty about the number of possible outcomes is a real and challenging statistical problem. See for example Inference for the binomial N parameter: A hierarchical Bayes approach(abstract)(full paper pdf) by Adrian Raftery. Raftery’s prior for the number of outcomes is 1/N, but you can’t use that for coherent betting.
I think there’s also the question of inferring the included name space and possibility space from the questions asked.
If he asks you about html color #FF0000 (which is red) after asking you about red, do you change your probability? Assuming he’s using 12 color words because he used ‘red’ is arbitrary.
Even with defined and distinct color terms, the question is, what of those colors are actual possibilities (colors in the jar) as opposed to logical possibilities (colors omega can name)
and I think THAT ties back to Elizer’s article about Job vs. Frodo.
I was mentally categorizing that as “Omega deliberately screwing with you” by using English strangely, but perhaps that was unmotivated of me. But this gets into a grand metaphysical discussion about where colors begin and end, and whether there is real vagueness around their borders, and a whole messy philosophy of language hissy fit about universals and tropes and subjectivity and other things that make you sound awfully silly if you argue about them in public. I ignored it because the idea of the post wasn’t about colors, it was about probabilities.
Personally, I think the intent has less to do with classifying colors strangely and more to do with finding a broader example where even less information is known. The misstep I think I took earlier had to do with assuming that the colors were just part of an example and the jar could theoretically hold items from an infinite set.
I get that when picking beads from the set of 12 colors it makes sense to guess that red will appear with a probability near 1⁄12. An infinite set, instead of 12, is interesting in terms of no information as well. As far as I can tell, there is no good argument for any particular member of the set. So, asking the question directly, what if the beads have integers printed on them? What am I supposed to do when Omega asks me about a particular number?
Unless you have a reason to believe that there is some constraint on what numbers could be used—if only a limited number of digits will fit on the bead, for example—your probability for each integer has to be infinitesimal.
You’re not allowed to do that. With a countably infinite set, your only option for priors that assign everything a number between 0 and 1 is to take a summable infinite series. (Exponential distributions, like that proposed by Peter above, are the most elegant for certain questions, but you can do p(n)=cn^{-2} or something else if you prefer to have slower decay of probabilities.)
In the case with colors rather than integers, a good prior on “first bead color, named in a form acceptable to Omega” would correspond to this: take this sort of distribution, starting with the most salient color names and working out from there, but being sure not to exceed 1 in total.
Of course, this is before Omega asks you anything. You then have to have some prior on Omega’s motivations, with respect to which you can update your initial prior when ve asks “Is it red?” And yes, you’ll be very metauncertain about both these priors… but you’ve got to pick something.
The point is that your probability for the “first” integers will not be infinitesimal. If you think that drops off too quickly, instead of 2 use 1+e or something. p(n) = e/(e+2) * (1+e)^(-|n|). And replace n with s(n) if you don’t like that ordering of integers. But regardless, there’s some N for which there is an n with |n|N such that p(n)/p(m) >> 1.
I wasn’t talking about limiting frequencies, so don’t ask me “how often?”
Would you bet $1 billion against my $1 that no number with absolute value smaller than 3^^^3 will come up? If not then you shouldn’t be assigning infinitesimal probability to those numbers.
I get the feeling that I am thinking about this incorrectly but am missing a key point. If someone out there can see it, please let me know.
I wasn’t talking about limiting frequencies, so don’t ask me “how often?”
Sorry.
If the set of possible options is all integers and Omega asks about a particular integer, why would the probability go up the smaller the number gets?
Would you bet $1 billion against my $1 that no number with absolute value smaller than 3^^^3 will come up? If not then you shouldn’t be assigning infinitesimal probability to those numbers.
Betting on ranges seems like a no brainer to me. If Omega comes and asks you to pick an integer and then asks me to bet on whether an object pulled from the jar will have an absolute value over or under that integer, I should always bet that the number will be higher than yours.
If I had a random number generator that could theoretically pull a random number from all integers, it seems weird to assume it will be small. As far as I know, such a random number generator is impossible. Assuming it is impossible, there must be a cap somewhere in the set of all integers. The catch is that we have no idea where this cap is. If you can write 3^^^3 I can write 3^^^3 + 1 which leads me to believe that no matter what number you pick, the cap will be significantly higher. As long as I can cover the costs of the bet, I should bet against you.
The math works like this:
Given the option to place $X against your $Y
That when you pick an integer Z
Omega will pull a number out of the jar that is greater than the absolute value of Z,
There is a way to express X / Y * Z + 1 and,
Assuming I am placing equal probabilities on each possible integer between 0 and X / Y * Z + 1,
I should always take the bet
A trivial example: If I bet $5 against your $1 and you pick the integer 100, I can easily imagine the number 501. 501⁄100 is greater than 5⁄1. I should take the bet.
The problem seems to be that I am placing equal probabilities on each possible integer while you favor numbers closer to 0. Favoring numbers like 1 or 2 makes a lot of sense if someone came up to me on the street with a bucket of balls with numbers printed on them. I would also consider the chances of pulling 1 to be much higher than 3^^^3.
So, perhaps, my misstep is thinking of Omega’s challenge as a purely theoretical puzzle and not associating it with the real world. In any case, I certainly do not want to give the impression that I think 3^^^3 is just as likely to appear as 42 in the real world. Of course, in the real world I wouldn’t bet on anything at all because I do not consider the information available to be useful in determining the correct action and I am ridiculously risk averse.
The problem seems to be that I am placing equal probabilities on each possible integer while you favor numbers closer to 0.
You are not doing so, since it is impossible. No such probability distribution exists. In fact you recognize this by saying there’s a cap somewhere out there, you just don’t know where. Well, this cap means that small numbers (smaller than the cap) have much, much higher probability (i.e. nonzero) than large numbers (those higher than the cap have zero probability).
Maybe this will serve as an intuition pump: suppose you’ve narrowed down your cap to just a few numbers. In fact, just N and 2N. You’ve given them each equal weight. Well, now p(1) = (1/N + 1/2N)/2 = 3⁄4 N, but p(N+k) = 1⁄4 N, and p(2N+k) = 0. The probability goes down as numbers get larger. Determine your priors over all the caps, compute the resulting distribution, and you’ll find p(n) eventually start to decrease.
(Edit) After rereading my own comment, I do not think much of anything in here make sense. Feel free to ignore it completely. I know what I was trying to say but failed miserably. Sorry.
But now you are playing semantics and are making artificial definitions on the types of beads in the jar. This is definitely not no information and somewhat demeans the original example. If we switched the example to balls with integers printed on them you would have no linguistic basis to say there are only twelve options. I am just assuming that this is a better example than the colored beads. If you specifically meant the article to use “no information” to exclude “linguistic hints” than I would be forced to agree with your conclusion. Relevant quotes from the original post:
But because you start with no information, it’s very hard to gather more.
Assuming you don’t think Omega is out to deliberately screw with you, you could say that the probability is .083 based on the fact that “red” is one of twelve basic color words in English.
But this .083 guess is as wrong as .5 in the numbered balls example. The 50⁄50 guess has nothing to do with “red” and everything to do with guessing correctly. I could translate it into the following statement with no qualms:
“Omega will pull a bead in the color of his choosing.”
If “color of his choosing” means red, okay. If it means blue, okay. I am not going to take one bet for each color because the color is unimportant until we see a bead come out of the jar.
Realistically, I would start at 0 because a bet with no information scares me, but the probability of “0“ is no more wrong than ”.5”. It just carries less risk.
You should not guess that the first bead has a 50% chance of being red, because if you do, you can have this conversation: [snip]
With the numbered balls example, anything but 0 is a foolish response because instead of red, blue, green … yellow it would be 1, 2, 3, 4 … NAN. But even still, “0“ is as wrong as ”.5” because we have no information.
(Off-topic) This conversation strangely reminds me of talking about Pascel’s Wager...
You should not guess that the first bead has a 50% chance of being red, because if you do, you can have this conversation:
Omega: What is the probability of the first bead being red as opposed to non-red?
You: Fifty-fifty.
Omega: So you would consider it more than fair if I offered you three dollars if the bead is red, and you paid me a dollar if it was non-red?
You: Sure, I’ll take that bet.
Omega: What is the probability of the first bead being blue as opposed to non-blue?
You: Fifty-fifty.
Omega: So you would consider it more than fair if I offered you three dollars if the bead is blue, and you paid me a dollar if it was non-blue?
You: Sure, I’ll take that bet.
(...and so on for ten more colors.)
Omega pulls out a red bead. He owes you three dollars, but you owe him eleven dollars. He wins.
You could have that conversation, but you don’t have to. The argument for assigning 50% to red is that it’s the only question Omega has asked you. There are several ways out of that. The first one is that the moment he offers you a 25% bet I would update to presume that 3:1 is not a positive e.v. bet, with a new number of perhaps 12.5% with a range of 0% to 25% with symmetric distribution. Similarly, if he offered me three to one that it wasn’t red, I would presume that it probably will be. On a similar note, when he asks about blue (even without any bets involved) I can’t see answering higher than 33.3%.
Contrast this with Alicorn watching this incident and offering me 3:1 after Omega asks my probability for red and I say 50%. I still have to update for Alicorn’s opinion, but I might or might not accept that bet.
The estimate should take into account the expectation of being asked further questions. The ignorance prior is applied to a model of observation. The model of observation expresses which questions you may be asked, and what structure will the dependencies between these possible observations have.
I do not know how related this is to your comment, but it made me think of another response to the Dutch book objection. (Am I using that term correctly?)
If Omega asks me about a red bead I can say 100%. If he then asks about a blue bead I can adjust my original estimate so that both red and blue are the same at 50⁄50. Every question asked is adding more information. If Omega asks about green beads all three answers get shifted to 1⁄3.
This translates into an example with numbered balls just fine. The more colors or numbers Omega asks about decreases the expected probability that any particular one of them will come out of the jar simply because the known space of colors and numbers is growing. Until Omega acknowledges that there could be a bead of that color or number there is no particular reason to assume that such a bead exists.
If the example was rewritten to simply say any type of object could be in the jar, this still makes sense. If Omega asks about a red bead, we say 100%. If Omega asks about a blue chair, both become 50%. The restriction of colors and numbers is our assumed knowledge and has nothing to do with the problem at hand. We can meta-game all we want, but it has nothing to do with what could be in the jar.
The state of the initial problem is this:
A red bead could be in the jar
After the second question:
A red bead could be in the jar
A green bead could be in the jar
I suppose it makes some sense to include an “other” category, but there is no knowledge of anything other than red and green beads. The question of probability implies that another may exist, but is that enough to assign it a probability?
Every question asked is adding more information. If Omega asks about green beads all three answers get shifted to 1⁄3.
I don’t think we should treat omega as adding (much) new information with each question. Omega is super intelligent, we should assume that he’s already went all the way down the rabbit hole of possible colors, including ones that our brains could process but our eyes don’t see. We’re not inferring anything about his state of mind because he’s only asking questions about red, green, and blue. A sequence of lilac turquoise turquoise lilac lilac says very much more about what’s in the jar than the two hundred color questions omega asked you beforehand.
Not every question Omega could ask would provide new information, but some certainly do. Suppose his follow-up questions were “What is the probability that the bead is transparent?”, “What is the probability that the bead is made of wood?” and “What is the probability that the bead is striped?”. It is very likely that your original probability distribution over colors implicitly set at least one of these answers to zero, but the fact that Omega has mentioned it as a possibility makes it considerably more likely.
If Omega asking if the bead could be striped changes you probability estimates, then you were either wrong before or wrong after (or likely both)
If omega tells you at the outset that the beads are all solid colors, then you should maintain your zero estimate that any are striped. If not, then you never should have had a zero estimate. He’s not giving you new information, he’s highlighting information you already had (or didn’t have.)
I don’t see any way to establish a reliable (non-anthropomorphic) chain of causality that connects there being red beads in the jars with Omega asking about red beads. He can ask about beads that aren’t there, and that couldn’t be there given the information he’s given you. When Omega offered to save x+1 billion people if the earth was less than 1 million years old, I don’t think anyone argued that his suggesting it should change our estimates.
There’s no need to, because probability is in the mind.
If you’re going to update based on what omega asks you then you must believe there is a connection that you have some information about.
If we don’t know anything about omega’s thought process or goals, then his questions tell us nothing.
I think our only disagreement is semantic.
If I initially divide the state space into solid colours, and then Omega asks if the bead could be striped, then I would say that’s a form of new information—specifically, information that my initial assumption about the nature of the state space was wrong. (It’s not information I can update on; I have to retrospectively change my priors.)
Apologies for the pointless diversion.
An ideal model of the real world must allow any miracle to happen, nothing should be logically prohibited.
Of note, I was operating under a bad assumption with regards to the original example. I assumed that the set was a finite but unknown set of colors or an infinite set of colors. In the former case, every question is giving a little information about the possible set. In the latter it really does not matter much.
Yes, this is true. Personally, I am still curious about what to do with the two hundred color questions.
Don’t think of probability as being mutable, as getting updated. Instead, consider a fixed comprehensive state space, that has a place on it for every possible future behavior, including the possible questions asked, possible pieces of evidence presented, possible actions you make. Assign a fixed probability measure to this state space.
Now, when you do observe something, this is information, an event, a subset on the global state space. This event selects an area on it, and encompasses some of the probability mass. The statements, or beliefs (such as “the ball #2 will be red”), that you update on this info, are probabilistic variables. A probabilistic variable is a function that maps the state space on a simpler domain, for example a binary discrete probabilistic variable is basically an event, a subset of the state space (that is, in some states, the ball #2 is indeed defined to be red, these states belong to the event of ball #2 being red).
Your info about the world retains only the part of the state space, and within that part of the state space, some portion of the probability mass goes to the event defining your statement, and some portion remains outside of it. The “updating” only happens when you focus on this info, as opposed to the whole state space.
If that picture is clear, you can try to step back to consider what kind of probability measure you’d assign to your state space, when its structure already encodes all possible future observations. If you are indifferent to a model, the assignment is going to be some kind of division into equal parts, according to the structure of state space.
IAWYC, but as pedagogy it’s about on the level of “How should you imagine a 7-dimensional torus? Just imagine an n-dimensional torus and let n go to 7.”
Eliezer’s post on priors explains the same idea more accessibly.
EDIT: Sorry, I didn’t notice you already linked it below.
What if Omega wants you to commit to a bet based on your probabilities at every step?
Or what if he just straight up asks you what color you want to guess the bead will be, without asking about any individual colors? (Then you’d probably be best served by switching to a language with fewer basic color words, but that aside...)
Than you are forced to bid 0 because you have to account for any further questions, which sounds similar to what Vladimir_Nesov said.
By the way, I think adding another restriction to your example to force it back into your specific response is not particularly meaningful. In the case where you do not have to commit to a bet at every step, does what I say make sense? If so, than what Vladimir_Nesov suggested seems to be on the right path with regards to your restrictions.
Switching languages is a semantic trick. If we are allowed to use any words to describe the bead we can just say “not-clear” because the space of “not-clear” covers what we generally mean by “color”. We may as well say “the bead will be a colored bead.” All of this breaks the assumed principle of no information.
If Omega wanted a particular color and forced us into actually answering the annoying question, we are completely off the path of probabilities and it does not matter what you answer as long as you picked a color. If Omega then asked us what the probability of that particular color coming out of the jar would be, the answer should be the same as if you picked any other color. This drops to zero unless you self-restrict by the number of colors you can personally remember.
MrHen, whatever strategy you’re employing here, it doesn’t sound like a strategy for arriving at the really truly correct answer, but some sort of clever set of verbal responses with a different purpose entirely. In real life, just because Omega asked if the bead is red simply does not mean there is probability 0 of it being green.
Mmm… I was not trying to employ a strategy with clever verbal responses. I thought I was arguing against that, actually, so I must be far from where I think I am.
I feel like I am trying to answer a completely different question than the one originally asked. Is the question:
Knowing nothing about what is in the jar except that its contents are divided by color as per our definition of “color”, what is the probability of a red bead being pulled?
Knowing nothing about what is in the jar, what is the probability of a red bead being pulled?
I admittedly assumed the latter even though the article used words closer to the former. Perhaps this was my mistake?
I would agree. I do think that Omega asking about a red bead implies nothing about the probability of it being green. What I am currently wondering is if the question implies anything about the probability of the bead being red. If Omega acknowledges that the bead could be red, does that give red a higher probability than green?
I suppose I instinctively would answer affirmatively. The reasoning is that “red” is now included in the jar’s potential outcomes while green has not been acknowledged yet. In other words, green doesn’t even have a probability. Strictly speaking, this makes little sense, so I must be misstepping somewhere. My hunches are pointing toward my disallowing green into the potential outcomes.
This does not mean that I refuse to think of green as a color, but that green is not automatically included in the jar’s potential outcomes just because Omega used the word “color”. Is this the verbal cleverness you were referring to?
(Switching thoughts) In terms of arriving at the really truly correct answer, it seems that a strategy that gets closer as more beads is what is desired. If no beads are revealed, what sort of strategy is possible? I think the answer to this revolves around my potential confusion of the original question.
I apologize if I am mudding things up and am way off base.
Is Omega privileging the hypothesis that the bead is red?:-)
Me: No, because you have more information than I do, and the fact that you would offer this bet is evidence that I should use to update my epistemic probabilities.
Well, Omega doesn’t really need the money. There’s no reason to believe he would balk at offering you a more-than-fair bet.
Well, in the case of Omega, I would at least suspect that he intends to demonstrate that I am vulnerable to a Dutch book, even though he doesn’t need the money.
If you meet an Omega, that is pretty good evidence that you are living in a simulation: specifically, you are being simulated inside a philosopher’s brain as a thought experiment.
Sorry, you haven’t convincingly demonstrated the wrongness of 50%. MrHen’s position seems to me quite natural and defensible, provided he picks a consistent prior. For example, I’d talk with Omega exactly as you described up to this point:
...Omega: What is the probability of the first bead being blue as opposed to non-blue?
Me: 25%.
You ask why 25%? My left foot said so… or maybe because Omega mentioned red first and blue second. C’mon Dutch book me.
I think that doing it this way assumes that Omega is deliberately screwing with you and will ask about colors in a way that is somehow germane to the likelihood. Assume he picked “red” to ask about first at random out of whatever colors the beads come in.
This new information gives me grounds to revise my estimates as Omega asks further questions, but I still don’t see how it demonstrates the wrongness of initially answering 50%.
The reason 50⁄50 is bad is because the beads in the jar come in no more than 12 colors and we have no reason to favor red over the other 11 colors.
Knowing there is a cap of 12 possible options, it makes intuitive sense to start by giving each color equal weights until more information appears. (Namely, whenever Omega starts pulling beads.)
We have a reason: Omega mentioned red.
I suppose the relevant question is now, “Does Omega mentioning red tell us anything about what is in the jar?” When we know the set of possible objects in the jar, it really tells us nothing new. If the set of possible objects is unknown, now we know red is a possibility and we can adjust accordingly.
The assumption here is that Omega is just randomly asking about something from the possible set of objects. Essentially, since Omega is admitting that red could be in the jar, we know red could be in the jar. In the 12 color scenario, we already know this. I do not think that Omega mentioning red should effect our guess.
All this arguing about priors eerily resembles scholastics, balancing angels on the head of a pin. Okay I get it, we read Omega’s Bible differently: unlike me, you see no symbolic significance in the mention of red. Riiiiight. Now how about an experiment?
Agreed. For what it is worth, I do see some significance in the mention of red, but cannot figure out why and do not see the significance in the 12 color example. This keeps setting off a red flag in my head because it seems inconsistent. Any help in figuring out why would be nifty.
In terms of an experiment, I would not bet at all if given the option. If I had to choose, I would choose whichever option costs less and right it off as a forced expense.
In English: If Omega said he had a dollar claiming the next bead would be red and asked me what I bet I would bet nothing. If I had to pick a non-zero number I would pick the smallest available.
But that doesn’t seem very interesting at all.
Then each time Omega mentions another color, it increases the expected number of colors the beads come in.
I think Alicorn is operating under a strict “12 colors of beads” idea based on what a color is or is not. As best as I can tell, the problem is essentially, “Given a finite set of bead colors in a jar, what is the probability of getting any particular color from a hidden mixture of beads?” The trickiness is that each color could have a different amount in the jar, not that there are any number of colors.
Alicorn answered elsewhere that when the jar has an infinite set of possible options the probability of any particular option would be infinitesimal.
If the number of possible outcomes is finite, fixed and known, but no other information is given, then there’s a unique correct prior: the maxentropy prior that gives equal weight to each possibility.
(Again, though, this is your prior before Omega says anything; you then have to update it as soon as ve speaks, given your prior on ver motivations in bringing up a particular color first. That part is trickier.)
How would you update given the following scenarios (this is assuming finite, fixed, known possible outcomes)?
Omega asks you for the probability of a red bead being chosen from the jar
Omega asks you for the probability of “any particular object” being chosen
Omega asks you to name an object from the set and then asks you for the probability of that object being chosen
I don’t think #2 or #3 give me any new relevant information, so I wouldn’t update. (Omega could be “messing with me” by incorporating my sense of salience of certain colors into the game, but this suspicion would be information for my prior, and I don’t think I learn anything new by being asked #3.)
I would incrementally increase my probability of red in case #1, and decrease the others evenly, but I can’t satisfy myself with the justification for this at the moment. The space of all minds is vast; and while it would make sense for several instrumental reasons to question first about a more common color, we’re assuming that Omega doesn’t need or want anything from this encounter.
In the real-life cases which this is meant to model, though, like having a psychologist doing a study in place of Omega, I can model their mind by mine and realize that there are more studies in which I’d ask about a color I know is likely to come up, than studies in which I’d pick a specific less-likely color, and so I should update p(red) positively.
But probably not all the way to 1⁄2.
To make consistent bets, we need a prior on the number of possible outcomes.
The fact that Omega is speaking English and uses the word “red” as opposed to “scarlet” or something is decent evidence that there are twelve colors in beadspace.
What happens if you’ve taken bets on twelve colors [ETA: that eat up all your probability] and then Omega asks you to name odds on a transparent bead?
I was operating under the assumption that clear is not a “solid color”.
Right you are. I didn’t read the original problem carefully enough…
Nevertheless, you can replace “transparent” with a surprising color like lilac, fuchsia, or, um, cyan to restore the effect. The point is that even decent evidence that there are twelve colors in beadspace doesn’t justify a probability distribution on the number of colors that places all of its mass at twelve.
The twelve basic colors are so called because they are not kinds of other colors. Lilac and fuchsia are kinds of purple (I guess you could argue that fuchsia is a kind of red, instead, but pretend you couldn’t), and cyan is a kind of blue. Even if you pull out a navy bead and then a cyan bead, they are both kinds of blue in English; in Russian, they would be different colors as unalike as pink and red.
So you’re arguing that by definition, the basic color words define a mutually exclusive and exhaustive set. But there are colors near cyan which are not easy to categorize—the fairest description would be blue-green. In the least convenient world, when Omega asks you for odds on blue-green, you ask it if that color counts as blue and/or green, and it replies, “Neither; I treat blue-green as distinct from blue and green.” Then what do you do?
I was mentally categorizing that as “Omega deliberately screwing with you” by using English strangely, but perhaps that was unmotivated of me. But this gets into a grand metaphysical discussion about where colors begin and end, and whether there is real vagueness around their borders, and a whole messy philosophy of language hissy fit about universals and tropes and subjectivity and other things that make you sound awfully silly if you argue about them in public. I ignored it because the idea of the post wasn’t about colors, it was about probabilities.
That’s a shame, because uncertainty about the number of possible outcomes is a real and challenging statistical problem. See for example Inference for the binomial N parameter: A hierarchical Bayes approach (abstract)(full paper pdf) by Adrian Raftery. Raftery’s prior for the number of outcomes is 1/N, but you can’t use that for coherent betting.
I think there’s also the question of inferring the included name space and possibility space from the questions asked.
If he asks you about html color #FF0000 (which is red) after asking you about red, do you change your probability? Assuming he’s using 12 color words because he used ‘red’ is arbitrary.
Even with defined and distinct color terms, the question is, what of those colors are actual possibilities (colors in the jar) as opposed to logical possibilities (colors omega can name)
and I think THAT ties back to Elizer’s article about Job vs. Frodo.
Personally, I think the intent has less to do with classifying colors strangely and more to do with finding a broader example where even less information is known. The misstep I think I took earlier had to do with assuming that the colors were just part of an example and the jar could theoretically hold items from an infinite set.
I get that when picking beads from the set of 12 colors it makes sense to guess that red will appear with a probability near 1⁄12. An infinite set, instead of 12, is interesting in terms of no information as well. As far as I can tell, there is no good argument for any particular member of the set. So, asking the question directly, what if the beads have integers printed on them? What am I supposed to do when Omega asks me about a particular number?
Unless you have a reason to believe that there is some constraint on what numbers could be used—if only a limited number of digits will fit on the bead, for example—your probability for each integer has to be infinitesimal.
You’re not allowed to do that. With a countably infinite set, your only option for priors that assign everything a number between 0 and 1 is to take a summable infinite series. (Exponential distributions, like that proposed by Peter above, are the most elegant for certain questions, but you can do p(n)=cn^{-2} or something else if you prefer to have slower decay of probabilities.)
In the case with colors rather than integers, a good prior on “first bead color, named in a form acceptable to Omega” would correspond to this: take this sort of distribution, starting with the most salient color names and working out from there, but being sure not to exceed 1 in total.
Of course, this is before Omega asks you anything. You then have to have some prior on Omega’s motivations, with respect to which you can update your initial prior when ve asks “Is it red?” And yes, you’ll be very metauncertain about both these priors… but you’ve got to pick something.
I am happy with that explanation. Thanks.
Why not, say, p(n) = (1/3) * 2^(-|n|)?
If p(n) = (1/3) * 2^(-|n|), then:
p(1) = (1/3) * 2^(-1) = 0.166666667
p(86) = (1 / 3) * (2^(-86)) = 4.30823236 × 10^(-27)
p(1 000 000) = (1/3) * 2^(-1 000 000) = Lower than Google’s calculator lets me go
Are you willing to bet that 1 is going to happen that much more often than 1,000,000?
The point is that your probability for the “first” integers will not be infinitesimal. If you think that drops off too quickly, instead of 2 use 1+e or something. p(n) = e/(e+2) * (1+e)^(-|n|). And replace n with s(n) if you don’t like that ordering of integers. But regardless, there’s some N for which there is an n with |n|N such that p(n)/p(m) >> 1.
I wasn’t talking about limiting frequencies, so don’t ask me “how often?”
Would you bet $1 billion against my $1 that no number with absolute value smaller than 3^^^3 will come up? If not then you shouldn’t be assigning infinitesimal probability to those numbers.
I get the feeling that I am thinking about this incorrectly but am missing a key point. If someone out there can see it, please let me know.
Sorry.
If the set of possible options is all integers and Omega asks about a particular integer, why would the probability go up the smaller the number gets?
Betting on ranges seems like a no brainer to me. If Omega comes and asks you to pick an integer and then asks me to bet on whether an object pulled from the jar will have an absolute value over or under that integer, I should always bet that the number will be higher than yours.
If I had a random number generator that could theoretically pull a random number from all integers, it seems weird to assume it will be small. As far as I know, such a random number generator is impossible. Assuming it is impossible, there must be a cap somewhere in the set of all integers. The catch is that we have no idea where this cap is. If you can write 3^^^3 I can write 3^^^3 + 1 which leads me to believe that no matter what number you pick, the cap will be significantly higher. As long as I can cover the costs of the bet, I should bet against you.
The math works like this:
Given the option to place $X against your $Y
That when you pick an integer Z
Omega will pull a number out of the jar that is greater than the absolute value of Z,
There is a way to express X / Y * Z + 1 and,
Assuming I am placing equal probabilities on each possible integer between 0 and X / Y * Z + 1,
I should always take the bet
A trivial example: If I bet $5 against your $1 and you pick the integer 100, I can easily imagine the number 501. 501⁄100 is greater than 5⁄1. I should take the bet.
The problem seems to be that I am placing equal probabilities on each possible integer while you favor numbers closer to 0. Favoring numbers like 1 or 2 makes a lot of sense if someone came up to me on the street with a bucket of balls with numbers printed on them. I would also consider the chances of pulling 1 to be much higher than 3^^^3.
So, perhaps, my misstep is thinking of Omega’s challenge as a purely theoretical puzzle and not associating it with the real world. In any case, I certainly do not want to give the impression that I think 3^^^3 is just as likely to appear as 42 in the real world. Of course, in the real world I wouldn’t bet on anything at all because I do not consider the information available to be useful in determining the correct action and I am ridiculously risk averse.
To dispel this confusion, you should read on algorithmic information theory.
Is there a good place to start online? Can I just Google “algorithmic information theory”?
Google first and ask questions later. ;-)
You are not doing so, since it is impossible. No such probability distribution exists. In fact you recognize this by saying there’s a cap somewhere out there, you just don’t know where. Well, this cap means that small numbers (smaller than the cap) have much, much higher probability (i.e. nonzero) than large numbers (those higher than the cap have zero probability).
Maybe this will serve as an intuition pump: suppose you’ve narrowed down your cap to just a few numbers. In fact, just N and 2N. You’ve given them each equal weight. Well, now p(1) = (1/N + 1/2N)/2 = 3⁄4 N, but p(N+k) = 1⁄4 N, and p(2N+k) = 0. The probability goes down as numbers get larger. Determine your priors over all the caps, compute the resulting distribution, and you’ll find p(n) eventually start to decrease.
(Edit) After rereading my own comment, I do not think much of anything in here make sense. Feel free to ignore it completely. I know what I was trying to say but failed miserably. Sorry.
But now you are playing semantics and are making artificial definitions on the types of beads in the jar. This is definitely not no information and somewhat demeans the original example. If we switched the example to balls with integers printed on them you would have no linguistic basis to say there are only twelve options. I am just assuming that this is a better example than the colored beads. If you specifically meant the article to use “no information” to exclude “linguistic hints” than I would be forced to agree with your conclusion. Relevant quotes from the original post:
But this .083 guess is as wrong as .5 in the numbered balls example. The 50⁄50 guess has nothing to do with “red” and everything to do with guessing correctly. I could translate it into the following statement with no qualms:
“Omega will pull a bead in the color of his choosing.”
If “color of his choosing” means red, okay. If it means blue, okay. I am not going to take one bet for each color because the color is unimportant until we see a bead come out of the jar.
Realistically, I would start at 0 because a bet with no information scares me, but the probability of “0“ is no more wrong than ”.5”. It just carries less risk.
With the numbered balls example, anything but 0 is a foolish response because instead of red, blue, green … yellow it would be 1, 2, 3, 4 … NAN. But even still, “0“ is as wrong as ”.5” because we have no information.
(Off-topic) This conversation strangely reminds me of talking about Pascel’s Wager...