I was pretty skeptical about your Bead Jar post, but now have changed my mind. This stuff is interesting, even disturbing. In the bead jar game it seems that one should assign tiny credence to "cerulean" and thus huge credence to "non-cerulean", but not be surprised when "non-cerulean" fails to occur. Do you have some kind of general theory of when surprise occurs or should occur?
I'm working on it. Clearly, "none of the above" situations - or the latter "Bingo" case - can rightly yield surprise.
Perhaps surprise is warranted when a well-supported model, rather than a well-calibrated probability, is disconfirmed. That doesn't explain why we should be surprised about a personal friend winning the lottery, though. That seems to be surprising solely because of astronomically low odds and the specialness of the outcome.
I think surprise might have to do with the difference between your expected and your actual Bayes score.
And lucky outcome could be defined by the difference between your previous expectation and updated expectation for the actual prize. But in both cases, I think you'd need to work with something like "knowledge about prior reviewed in light of new evidence" (reviewed knowledge about prior, not updated prior=posterior), compared with "knowledge about prior before that".
Then I don't understand why we'd be surprised to see a fair coin fall heads ten times in a row.
Because you assign the all-heads sequence a probability significantly higher than 2^-10, so your Bayes score is higher than you expected. Surprise!
Edit: I didn't notice that you said the coin is fair. Well, I'll bite the bullet and claim that if you really assign a probability of 1 to the coin being fair, then you won't feel surprised no matter how many times it comes up heads.
Agree. In practice, I'd bet that our pattern-seeking minds really do put more weight on simple fixed-coin hypotheses than we're consciously aware of; after only three heads in a row, such a hypothesis would pop into my head (though I'd consciously dismiss it), and after three or four more heads, I'd start to consciously consider it.
I wonder if the definition of "surprise" isn't a problem here - we may need two distinct words. The amount of information added to your knowledge of the universe is large when a low-probability event is observed. This is "surprising" in the sense of number of bits it takes to encode.
It's not "surprising" in the human emotional sense, because you've already aggregated a number of probabilities, and this is a common result.
Likewise, Omega drawing a cerulean bead is informative, but not surprising. Drawing the Battleship Potemkin is surprising, because you've (incorrectly) assigned it a zero probability.
The amount of information added to your knowledge of the universe is large when a low-probability event is observed. This is "surprising" in the sense of number of bits it takes to encode.
I don't think it necessarily takes a lot of bits to encode low-probability events. If I take out the ten of diamonds and the ace of diamonds and have you pick one of the two, the probability of ◊10 is 50%; if I leave all the cards in the deck, the probability of drawing ◊10 is 1/52, but it doesn't take more space to write ◊10 depending on whence the card came.
Drawing the Battleship Potemkin out of Omega's jar would be surprising because it messes with the definition of Omega, who said the jar contained solid-colored beads. A boat (or a film, I'm not sure which you meant), which is not a bead, disconfirms the model of Omega. (Or the model of oneself as a an agent who can remember things Omega says.)
Let me explain what Dagon meant, using your example. The total information required to select 1 card out of 52, e.g. ◊10, is about 6 bits (think of it as 6 divisions in half). In the first case you receive 5 of those bits when you're told what the two cards are, and 1 more bit when you actually draw the card. Only that last bit depends on the random event. In the second case you receive all 6 at once, so all 6 depend on the random event.
ETA: I didn't downvote you.
That doesn't explain why we should be surprised about a personal friend winning the lottery, though. That seems to be surprising solely because of astronomically low odds and the specialness of the outcome.
And the outcome is special only because the set of people categorized as "personal friend" is determined before the lottery winner is announced.
If you draw a card at random from a shuffled deck, whatever card you get had a 1/52 chance of being selected; this is only surprising if you predicted in advance that it would be that specific card.
Far trickier is how to determine "surprisingness" in cases where the space of possible outcomes is partially or completely unknown.
If you draw a card at random from a shuffled deck, whatever card you get had a 1/52 chance of being selected; this is only surprising if you predicted in advance that it would be that specific card.
So clearly, if I write another post on this, I'll have to call it "Was Your Card the Ten of Diamonds?"
Maybe it does explain why we're surprised about a personal friend winning the lottery - if we identify the "well-supported model" we were relying on.
Note, it need not be a model which is well-supported in terms of epistemic rationality. Merely that the model has been instrumentally useful: i.e. "my personal friend won't become incredibly wealthy without warning."
Alternatively, maybe it is worth considering different types of surprise which have some things in common but some differences.
IIRC, the conclusion was that "surprise" is when some low-probability complex hypothesis suddenly rises to prominence. Thus it's not something that describes one of 100000 same-probability events happening, but something that describes one winning a lottery 10 times in a row, or an old lame horse winning the race, in which case you start suspecting that something is going on. If a low-probability event doesn't give a hint that something unexpected is going on, there is no surprise.
The emotion of surprise itself is possibly an adaptation that tells the brain to pay attention, to try to figure out what that new unexpected phenomenon might be and what else that entails.
Picometer nitpick, for accuracy:
你有鼻子, as phrased, is not a question and thus ironically even more bewildering, not that someone who couldn't understand the utterance would be able to determine that. To phrase it as a question you need a different form; one of
你有鼻子吗？ 你有没有鼻子？ or 你有鼻子，对不对？
would work. The first is a simple question. The second leaves a bit more credence to the possibility you don't have a nose. The third probably is trying to imply that if you don't agree then you're foolish.
Maybe I'm missing some nuances here, but couldn't we just say we're surprised when "we are presented with a new degree of freedom in a (stochastic or deterministic) model we previously had about the world"?
Take the 52 cards example. We expect to see a number and a suit on the card. If we're then presented with a card showing an unfamiliar number or suit (or any other gibberish, then our model has been falsified). There was one more possible outcome (degree of freedom) that we were previously unaware of.
I think it's because our minds unconsciously assign substantial weight to a number of hypotheses we'd consciously conclude are silly, along the lines of "what if my friend won the lottery this time", along with pattern-seeking hardware that makes us place too much weight on fixed-coin hypotheses, etc.
In other words, we've subconsciously singled out a small number of outcomes to keep an eye on, despite our conscious belief that these should represent a vanishing fraction of the probability mass. Thus the potential for surprise.
Were we Bayesians instead of Godshatter (and if we somehow had a prior with extremely strong likelihood that this lottery was genuinely fair), then our friend winning might not surprise us.
I'm interested to see what happens to your Chinese. I noticed that some non-ASCII entities in in a top-level comment looked okay right after I posted, but were mangled later (without my doing any further edits).
I see five question marks. (Which I thought was intended until I got to the "if you can read Chinese" part.)
I just checked the source of my post against my saved local copy. The
entities that were given by name were converted to hexadecimal and look
fine (’ became “ ) but the
entities that were in decimal were mangled into multiple hexadecimal
entities (↩ became ↊).
I didn't have any that were originally in hex, and I don't know if this
difference is the reason it was mangled. It looks like all of your
Chinese characters are currently hex entities (the answer in the footnote is 有).
In Bead Jar Guesses, I made a slightly clumsy attempt at carving out a kind of guess based on so little information that even a rationally-supposed, very small probability of some outcome doesn't confer a commensurate level of surprise when that outcome occurs. Here are several categories of probability assignment (including a re-statement of the bead jar thing) that I think might be worth considering separately. (I'm open to changing their names if other people have better ideas.)
Bewilderment: You don't even have enough information to understand the question. What is your probability that any given shren is a darkling? What is your probability that Safaitic is sometimes recorded in boustrophedon? What is your probability that 你有鼻子? (Ignore question 1 if you have read Elcenia, especially if you've seen more than is currently published; ignore question 2 if you know what either of those funny words mean; ignore question 3 if you can read Chinese.) In this case, you might find yourself in a situation where you have to make a guess, but even if you were then told the answer it wouldn't tell you much in the absence of further context.1 You would have no reason to be surprised by such an answer, no matter what probability you'd assigned.
Bead Jar: You understand the question, but have no information about anything that causally interacts with the answer. To guess, you have to grasp at the flimsiest of straws in the wording of the question and the motivations of the asker, or much broader beliefs about the general kind of question. What is your probability that Omega will pull out a red bead? What's your probability that I'm making the peace sign as I type this question with the other hand? What's your probability that the fruit on the tree in my best friend's backyard is delicious? Like Bewilderment questions, Bead Jar guesses come with no significant chance of surprise. Even if you have a tiny probability that the bead is lilac, it should not surprise you.
Bingo: You understand the question and you know something about what causes the answer, but the mechanism by which those conditions come about is known to be random (in a practical epistemic sense, not necessarily in the sense of being physically undetermined). You can have an excellent, well-calibrated probability. Here, there are two variants: one where the outcomes have mostly commensurate likelihood (the probability that you'll draw any given card from a deck) or one where the outcomes have a variety of probabilities (like the probability that you draw a card with a skull, or one with a star). You shouldn't be surprised no matter what happens in the first case (unless the outcome is somehow special to you - be surprised if a personal friend of yours wins the lottery!), but in the second case, surprise might be warranted if something especially unlikely happens.
1About 5/6 of shrens are darklings, depending on population fluctuations; Safaitic is indeed sometimes recorded in boustrophedon; and 有 (I hope).