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 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.
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).
That’s not true at all. Before I’m dealt a bridge hand, my probability assignment for getting the hand J♠, 8♣, 6♠, Q♡, 5♣, Q♢, Q♣, 5♡, 3♡, J♣, J♡, 2♡, 7♢ in that order would be one in 3,954,242,643,911,239,680,000. But I wouldn’t be the least bit surprised to get it.
In the terminology of statistical mechanics, I guess surprise isn’t caused by low-probability microstates ― it’s caused by low-probability macrostates. (I’d have been very surprised if that were a full suit in order, despite the fact that a priori that has the same probability.) What you define as a macrostate is to some extent arbitrary. In the case of bridge, you’d probably divide up hands into classes based on their utility in bridge, and be surprised only if you get an unlikely type of hand.
In this case, I’d probably divide the outcomes up into macrostates like “red”, “some other bright color like green or blue”, “some other common color like brown”, “a weird color like grayish-pink”, and “something other than a solid-colored ball, or something I failed to even think of”. Each macrostate would have a pretty high probability (including the last: who knows what Omega’s up to?), so I wouldn’t be surprised at any outcome.
This is an off-the-cuff analysis, and maybe I’m missing something, but the idea that any low-probability event should be surprising certainly can’t be correct.
Thank you, my mistake. I don’t understand ‘surprise’.
Let’s see… It looks like ‘surprise’ is something about promoting a new theory about the structure of environment that was previously dormant, forcing you to drop many cached assumptions. For example, if (surprise, surprise...) you win a lottery, you may promote a previously dormant theory that you are on a holodeck. If you are surprised by observing 1000 equal quantum coinflips (replicated under some conditions, with apparatus not to blame), you may need to reconsider the theory of physics. If you experience surprising luck in a game of dice, you start considering the possibility that dice are weighted.
… 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.)
“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.
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.
That’s not true at all. Before I’m dealt a bridge hand, my probability assignment for getting the hand J♠, 8♣, 6♠, Q♡, 5♣, Q♢, Q♣, 5♡, 3♡, J♣, J♡, 2♡, 7♢ in that order would be one in 3,954,242,643,911,239,680,000. But I wouldn’t be the least bit surprised to get it.
In the terminology of statistical mechanics, I guess surprise isn’t caused by low-probability microstates ― it’s caused by low-probability macrostates. (I’d have been very surprised if that were a full suit in order, despite the fact that a priori that has the same probability.) What you define as a macrostate is to some extent arbitrary. In the case of bridge, you’d probably divide up hands into classes based on their utility in bridge, and be surprised only if you get an unlikely type of hand.
In this case, I’d probably divide the outcomes up into macrostates like “red”, “some other bright color like green or blue”, “some other common color like brown”, “a weird color like grayish-pink”, and “something other than a solid-colored ball, or something I failed to even think of”. Each macrostate would have a pretty high probability (including the last: who knows what Omega’s up to?), so I wouldn’t be surprised at any outcome.
This is an off-the-cuff analysis, and maybe I’m missing something, but the idea that any low-probability event should be surprising certainly can’t be correct.
Thank you, my mistake. I don’t understand ‘surprise’.
Let’s see… It looks like ‘surprise’ is something about promoting a new theory about the structure of environment that was previously dormant, forcing you to drop many cached assumptions. For example, if (surprise, surprise...) you win a lottery, you may promote a previously dormant theory that you are on a holodeck. If you are surprised by observing 1000 equal quantum coinflips (replicated under some conditions, with apparatus not to blame), you may need to reconsider the theory of physics. If you experience surprising luck in a game of dice, you start considering the possibility that dice are weighted.
… 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.)
“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.
Well, maybe it isn’t, but it should.