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jimmy comments on Bead Jar Guesses - Less Wrong
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"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.
Well, I suppose you do have information inasmuch as you know what colors are. But if your probability for red is .5, on the basis of knowing that it's a color alone, then you have to have the same probabilities for blue and yellow and green and brown and so forth if Omega asks for those too, and you can be Dutch booked like crazy.
If you can be Dutch booked about probabilities asked in close sequence, where you gain no new information except that a question was asked, I'd think that reflects a considerable failure of rationality. There are grounds to reject "temporal Dutch book arguments", but this isn't one; the time and the "new information" should both be negligible.
To put it differently, if you have no information about what beads are in the jar, then you have even less information about why Omega wants to know your probabilities for the sorts of beads in the jar. Omega is a weird dude. Omega asking you a question does not mean what it means when a human asks you the same question.
And it's always .5, I hope.
Your probability of updating downwards should be (more or less; not exactly) equal to one minus your original probability, i.e. if your original probability is .25, your probability of updating downwards should be around .75. This is obvious, since if there is a one in four chance that the thing is so, there is a three out of four chance that you will find out that it is not so, when you find out whether it is so or not.
Conservation of expected evidence doesn't mean that the chance of updating upwards is equal to the chance of updating downwards. It also takes into account the quantity of the change; i.e. my probability is .25, and I update upwards, I will have to update three times as much as if I had updated downwards.
You're right. Thanks.
What if you know jar A is 80% red and jar B is 0% red, and you know you're looking at one of them, and your confidence that it's A is 0.625? Then you have probability 0.5 that a bead chosen from the jar in front of you is red, but will update upwards with probability 0.625 if you're given the information of which jar you're looking at.
My comment assigns to a probability to updating upwards or downwards in a generic way when new information is given; your comment calculates based on "if you're given the information of which jar you're looking at", which is more concrete. You could also be given other information which would make it more likely you're looking at B.
No, it's not. (You either win the lottery, or you don't.)
Excuse me for making such a minor point, but I don't think we have to give the same probability for each color. We have to guess at Omega's motivation before we can guess at the distribution of bead colors in the jar. Do we have previous knowledge of Omegas? How about Omegas bearing bead filled jars?
I was assuming that you have never met an Omega, much less one bearing a bead jar, and that you know all the standard facts about Omega (e.g. what he says is true, etc.)
I think I would agree partially with both of you. If I assume that there is no information at all .5 is a good choice. Once a bead of any color is pulled out, I can start making guesses on a potential number of beads in the jar from the relative volumes of the jar and the bead, so if I know that there is a finite number of potential colors, I might take a guess as to what the probability of any particular color distribution is. Once a red bead is pulled, I might adjust probability that Omega is not screwing with me etc.