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Alicorn comments on Bead Jar Guesses - Less Wrong
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Comments (127)
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:
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 and m with |m|>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.
(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...