"We'll bypass the novice mistake of calling it .5, of course; just because the options are binary (red or non-red) doesn't make them equally likely. It's not like you have any information."
Well, if you truly had no information, 0.5 would be the correct (entropy maximizing given constraints) bet. If you have no information you can call it "A or !A" or "!B or B" and it sounds the same- you can't say one is more likely.
By assigning a different probability, you're saying that you have information that makes the word "red" means something to you, and that it's less likely than half (say because there are 11 other "colors").
Likewise, if I say how likely is A and how likely is !A? You have to say 0.5. If A turns out to be "I'm gonna win the lottery tomorrow" then you can update and P goes to near zero. You didn't screw up though, since It could have just as easily been "I won't win the lottery tomorrow". If you don't think that it's just as likely, then that is information.
When you hear people saying "winning the lottery is 50/50 because either you win or you don't", their error isn't that they "naively" predict 0.5 in total absence of information. Their problem is that they don't update on the information that they do have.
I think more of our beliefs are bead jar guesses than we realize, but because of assorted insidious psychological tendencies, we don't recognize that and we hold onto them tighter than baseless suppositions deserve. [said Alicorn in the original]
If someone wants to do the work of linking the fairly abstract discussion here to how we think about making decisions in the real world, I think we would all benefit greatly.
We're to assume here that Omega, being a strange sort of deity, has selected the question "Is the bead red?" via some process that has no expected correlation whatsoever to the actual color. Correct?
I've just started reading Jaynes on prior formation, and I'd love to see more posts here on the topic. Maybe I'll write one if I ever have the chance to get some reading done.
As far as this problem goes, I agree we have some information about other colors. I want to know what Omega counts as "red" though, because that will go a long way in determining what sort of prior we'd assign.
Based on my limited understanding of physics, if we assume the bead only reflects a single wavelength, then it would be red if the wavelength were between 620 and 750...
It's not like you have any information. Assuming you don't think Omega is out to deliberately screw with you
These are contradictory assumptions. I have no information. I have no idea what Omega is out to do. The whole point of invoking Omega is to obliterate meaningful priors. I'm with byrnema here, probabilities are a tool for making maximally effective use of information. Without any such information, the only correct answer to "what is your probability" is "I don't have one".
Interestingly, figuring out the answers to questions of this kind, basically about prior, we are dealing with issues similar to those in elicitation of human values. In both cases, the answer is hidden in our minds, never in explicit and consistent form, with no hope of constructing a precise model that will give the answer. The only way to approximate the solution is to consider arguments for and against, consider relared situations, think, and listen to your inner voice, to intuitive response that says that it's proper to save a child, and you agree, tha...
Alicorn, I think it'd be appropriate to add the following link at the beginning of the article:
Related to: Priors as Mathematical Objects.
It also kinda answers your questions.
Even if Omega had asked about the bead being lilac, and you'd dutifully given a tiny probability, it would not have surprised you to see a lilac bead emerge from the jar.
I see this conclusion as a mistake: being surprised is a way of translating between intuition and explicit probability estimates. If you are not surprised, you should assign high enough probability, and otherw...
But because you start with no information, it's very hard to gather more. Suppose Omega reaches into the jar and pulls out a red bead. Does your probability that the second bead will be red go up... down... [or] stay the same...?
My intuition here is to start with an uninformative prior over possible bead-generating mechanisms. (You still have the problem of how to divide up the state space, but that's nothing new.) If a red bead comes out first, I update the probabilities that I assign to each mechanism and proceed from there.
Where exactly that leads ...
For me, the Omega problem described in the post presents the following conundrum: what is a probability in the limit of no information?
Suppose we employ a pragmatic perspective: the "probability of an event", as a mathematical object, is a tool that is used to summarize information about the past and/or future occurrence of that event. In the limit of no information, using the pragmatic view, there is no justification for assigning a probability not because we don't know what it is, but because it has no use in summarizing information.
If you do...
This post confused me enormously. I thought I must be missing something, but reading over the comments, this seems to be true for virtually all readers.
What exactly do you mean by "bead jar guess"? "Surprise"? "Actual probability"? Are you making a new point or explaining something existing? Are you purposely being obscure "to make us think"?
I propose replacing this entire post with the following text:
Hey everybody! Read E.T. Jaynes's Probability Theory: The Logic Of Science!
Thinking about how an Occamian learner like AIXI would approach the problem, it would probably start from the simplest domain theory "beads have a color, red is a color I've heard mentioned, therefore all beads are red", p=1. If the first bead was grey, it would switch to "all beads are grey", p=0. The second bead is red, "half and half", p = 0.5, and so on, ratcheting up theories from the simplest first.
I think this post could have been more formally worded. It draws a distinction between two types of probability assignment, but the only practical difference given is that you'd be surprised if you're wrong in one case but not the other. My initial thought was just that surprise is an irrational thing that should be disregarded ― there's no term for "how surprised I was" in Bayes' Theorem.
But let's rephrase the problem a bit. You've made your probability assignments based on Omega's question: say 1/12 for each color. Now consider another situ...
I'd say like the second example, most of our beliefs are bead jar guesses informed by untrustworthy informants. Namely our parents and other adults around when we were young.
Do you see this as being sort of like Jimmy's metauncertainty?
Also, if Omega pulled out a bead and then asked you about the next one, the Rule of Succession would be a good place to start making guesses.
Omega asking you that question, "What's the probability that the bead will be red" is itself information about the beads - Omega is more likely to ask that question in cases where the color is relevant.
To do things properly, you could suppose that Omega is inspiring himself from existing bead-color-guessing logic puzzles (there are plenty of examples of those in human history), and you could assign probabilities to each type of guessing games, noting how many types of beads there are etc. You can also collect statistics about which combinations o...
Apparently, the term you are searching for is "Second Order Probability".
See here for a paper: www.dodccrp.org/events/2000_CCRTS/html/pdf_papers/Track_4/124.pdf
I've just started reading Jaynes on prior formation, and I'd love to see more posts here on the topic. Maybe I'll write one if I ever have the chance to get some reading done.
As far as this problem goes, I agree we have some information about other colors. I want to know what Omega counts as "red" though, because that will go a long way in determining what sort of prior we'd assign.
Based on my limited understanding of physics, if we assume the bead only reflects a single wavelength, then it would be red if the wavelength were between 620 and 750...
I'm sure not sure I understand the point of this post. Are you saying that guesses without any information are inherently unfounded?
How would guessing 50% on the first pull be any worse since, by definition of the problem, you have no information? As soon as you have seen one bead, however, you have perfect historical information which is better than none.
Assuming that Omega is picking at random, it makes sense to me to simply pick a random percentage on the first pull and then swing to 0% or 100% once you see a bead. Update again on the second bead to ...
In my opinion, a more interesting question is what game Omega can devise in which revealing your probability estimate is part of the winning strategy. If he asks you to name even odds and bet with him, for example, you could name ridiculous odds if you wanted more money. The only thing I can think of is for Omega to pour out part of the jar and reward you depending on your deviation from the correct percentage.
Let's say Omega turns up and sets you a puzzle, since this seems to be what Omega does in his spare time. He has with him an opaque jar, which he says contains some solid-colored beads, and he's going to draw one bead out of the jar. He would like to know what your probability is that the bead will be red.
Well, now there is an interesting question. We'll bypass the novice mistake of calling it .5, of course; just because the options are binary (red or non-red) doesn't make them equally likely. It's not like you have any information. 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. (If he had asked for the probability that the bead would be lilac, you'd be in a bit more trouble.) If you were obliged to make a bet that the bead is red, you would probably take the most conservative bet available (even if you're still assuming Omega isn't deliberately screwing with you), but .083 sounds okay.
But because you start with no information, it's very hard to gather more. Suppose Omega reaches into the jar and pulls out a red bead. Does your probability that the second bead will be red go up (obviously the beads come in red)? Does it go down (that might have been the only one, and however many red beads there were before, there are fewer now)? Does it stay the same (the beads are all - as far as you know - independent of one another; removing this one bead has an effect on the actual probabilities of what the next one will be, but it can't affect your epistemic probability)? What if he pulled out a gray bead first, instead of a red one? How many beads would he have to pull, and in what colors, for you to start making confident predictions?
So that's one kind of probability: the bead jar guess. It has a basis, but it's a terribly flimsy one, and guessing right (or wrong) doesn't help much to confirm or disconfirm the guess. Even if Omega had asked about the bead being lilac, and you'd dutifully given a tiny probability, it would not have surprised you to see a lilac bead emerge from the jar.
A non-bead-jar-guess probability yields surprise when it turns out to be true even if it's just the same size. Say your probability for lilac was .003. That's tiny. If you had a probability of .003 that it would rain on a particular day, you would be right to be astonished if you turned out to need the umbrella you left at home.
Bead jar guesses vacillate more easily. Although in the case of the bead jar, you are in an extremely disadvantageous position when it comes to getting more information, we can fix that: somebody who says she's peeked into the jar says all the beads in the jar are red. Just like that, you'll discard the .083 and swap it for a solid .99 (adjusted as you like for the possibility that she is lying or can't see well). It would take considerable evidence to move a probability that far if it were not a wild guess, not just a single person's say-so, but that's all you've got. Then Omega pulling out a bead can give you information: the minute he pulls out the gray bead you know you can't rely on your informant, at least not completely. You can start making decent inferences.
I think more of our beliefs are bead jar guesses than we realize, but because of assorted insidious psychological tendencies, we don't recognize that and we hold onto them tighter than baseless suppositions deserve.