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Cyan comments on Bead Jar Guesses - Less Wrong
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Comments (127)
Alicorn, I think it'd be appropriate to add the following link at the beginning of the article:
It also kinda answers your questions.
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 otherwise if you assign tiny probability, you should be surprised (modulo known mistakes in either representation).
Predicting the second bead given the color of the first one can also be expressed as probability estimates for joint observations, made before you observe the color of the first bead. What is the probability that you'll see two reds? That you'll see a red followed by a non-red? Non-red following by a red? Two non-reds? Then crunch the numbers through the definition of conditional probability/Bayes' theorem.
... but it isn't, because the degree of surprise doesn't just depend on the raw probability, but also only the number of other possible outcomes under consideration. That Omega uses the term "lilac" may reasonably be taken as evidence that the space of color outcomes should be treated as finely divided.
ETA: I guess the mistake is in comparing feelings of surprise across outcomes with the same probability embedded in event spaces with different cardinalities.
If Omega asked me the probability of the next bead being lilac, I would be surprised to if the next bead actually was lilac, in a way I would not be surprised to find the bead is turquoise, an event to which I assign equal probability, but was not specifically considering prior to the draw, as any higher probability set of events which excludes drawing a turquoise bead would seem artificial. If the first two beads are the colors Omega asks me about, my leading theory would be that Omega will draw out a bead of which ever color he just brought up. (The first draw would cause me to consider this with roughly equal probability as maximum entropy.)
Well, maybe it isn't, but it should.
"doesn't just depend on the raw probability" - Correct. It also depends strongly on how reliable you think your estimate of the probability is. That is, your confidence interval.