"Nah, it's just an answer to your question.  Or at least, if it had some hugely important hidden meaning about how to avoid some dreadful Science!-related catastrophe, I didn't get it either, when it was emphasized to me as a kid."

"Anyways, I think we're drifting off-topic a bit."

"Returning to the main point, I suggested that if you were mounting a serious assault to conquer the Keltham-Environment, and you hadn't figured out yet how to get any result but 'YES', your next steps should include:  Measuring other things about me besides my 'YES' answer; measuring how much time I took to answer each input, as precisely as you could with your time-measuring instruments, and writing down all those results; trying to predict the time for new inputs you constructed."

"I don't know how much Probability you got off Asmodia, but can any of the researcher-candidates say what that advice truly means - why it makes sense - in terms of the Law of Probability?"

"Because it - removes twos; it destroys some of the probabilities, by exposing them to reality; it alters the ratio of the masses on each of the thousand sides of the scale, by evaporating the weight on them unevenly between them. The more we see that some types of questions seem to take longer for you to answer, the less likely it is that the time it takes you to answer a question is unrelated to the contents of the question."

"Every time you expose yourself to the reality of my 'YES' answer, if you weren't absolutely sure already that you'd get a YES answer, some of the probabilities evaporate.  That's already true.  What's different about adding the timing measurement, then?"

"How does the evaporation of probability change with measuring my timing?  How does it change with measuring my timing more precisely?  How does it change with trying to predict the timing for particular inputs instead of trying random inputs and passively observing the timed results?"

After receiving mental approval, Willa says, "I was really surprised by the first long pause after an input of very simple small numbers, then I tried bigger numbers and got a short pause, then I tried small numbers again and got another long pause. This seemed to sharply reduce the probability that whatever was being done to the numbers was a direct computation of some kind, and from there I realized the type of answers I'd been missing really fast."

"The leading theories I had were that long pauses meant difficult questions, long pauses were unrelated to the difficulty of the question, perhaps being randomly decided, and long pauses were a deliberate attempt to mislead me, 'trolling' in your parlance, in that order; I would have put roughly a 6:2:1 ratio on them, when I started, though in fact I did not think that explicitly. If I asked a question that I expected would be difficult and you answered slowly, that was evidence that either I was right it was difficult or you were trolling me, and also that - given that I believed it was a difficult question - that my first theory was right and my second and third theories were wrong.  If, on the other hand, I expected that it would be difficult and you answered quickly, that was evidence either that I was wrong about it being difficult or I was right and you were trolling me, and also that my leading theory was wrong and hence one of the others - or neither of the others - was correct."

"So that's the particular things that happened during your 'timing side-channel' experiments, and what you thought about them in Taldane.  The thing I'm asking for is more like -"

"Well, I called it Probability Sight at one point, by analogy to Arcane Sight, and Asmodia called it that too after she suddenly developed it one day."

"It's an abstract way of seeing things that would generalize to doing more Science later.  I'm trying to figure if there's some way to get it into people without either raising them as dath ilani, or dropping an artifact headband on them for two hours and inducing a permanent need for a cognitive prosthetic."

"Let me try -"

"So, a few days ago, when I was introducing the idea of lost-powers-of-2 as the 'truth-functional' relationship between a probability-claim, and reality, Carissa was briefly worried about how, if you could only ever lose 2s by playing that game, would that mean, people would be incentivized to avoid playing it?  And I was like, nothing bad directly happens to you when you lose a 2, it's just a relative measure of how well you're predicting, given that you're trying to predict at all."

"When you start trying to predict the timing, you lose more 2s, or 'bits' in Baseline.  The joint probability you assign to everything you see, goes down, because you see more.  But in the course of losing more 2s, you further narrow down what's probably true, and concentrate the remaining probability that hasn't been eliminated."

"This is not necessarily something that happens every time you manage to predict more and lose more 'bits'.  If that was always a productive activity, I could go on spinning a coin and seeing whether it came up showing the Queen's face, or the text on the reverse side, and lose one 'bit' every time.  The thing about the 'timing side-channel' is that some different hypotheses about what could be going on, lose different amounts of 2s; that's what makes it an informative experiment."

"That's why adding this new 'variable' to measure and predict, which makes you lose more 2s, is actually a great move, because the 2s are differentially coming from different hypotheses.  Figuring out which hypothesis is true, is what we care about, not how many 2s we lose along the way inside predictions we didn't stake lots of value on getting right."

"Why does it help to measure the times more precisely?"

"Because it's more specific?" 

"Can you be more meta-specific about exactly why it helps to be more specific?  In the language of Probability?"

"Because the more precise your claim is, the fewer twos you lose if you get it right and the more twos you lose if you get it wrong?

"I was asking about why it helps to have a more precise measurement, not to make more precise claims.  But maybe you're right and all we really need is the precise claim.  So we should measure my response times to within the nearest 6 seconds, but predict that my next response will take exactly 4.59823987 seconds.  This way we get all the benefits of a precise claim, but without the risk of losing lots of 2s by actually measuring.  Would you agree that this is the pathway of wisdom?  If not, why not?"

"More precise answers rule out more things? Like, if we're looking for a person, and we can measure where he is but only to within five hundred miles of precision, our measurement moves us towards all the theories that said he'd be in Cheliax, but can't tell the difference between theories that said he'd be in Egorian and theories that said he'd be in Ostenso."

"You're not actually gaining more information from the precise claim if you can't measure whether it was truer than a less-precise claim."

"This time it would've helped because if you kept taking data it would've been clear the pauses were slowly growing longer, on average, as more and more sequences were tried. After long enough, any explanation that didn't have a reason for the pauses to grow longer wouldn't predict that, and it would lose 2s."

"Because 'short' or 'long' is yes-or-no, but fractions-of-a-second are continuous; you can only get one bit out of a yes-or-no question, but many more out of a continuous one. The more data you have, the greater the updates you can make based on it."

"Having lots of precision can't possibly help - I mean, the difference between 4.59823987 seconds and 4.59823986 seconds - that difference isn't going to be related to anything -"

"I mean, it totally can help by the time you're Civilization, but you're not gonna get there for a while.  That's the kind of precise difference you'd run into if you were measuring the difference in exactly how fast time flows at the bottom of a mountain versus the top of a mountain."

"And, okay, let me finish responding to some of the previous things."

"You can't literally measure a continuous quantity and get a continuous answer, in fact, because that would correspond to infinite precision and an infinite number of 'bits' in the measurement.  It's also not true that you can only lose at most 1 'bit' on predicting something yes-or-no, for instance, you could predict YES with probability 63/64 and NO at 1/64 and see NO and lose 6 'bits'."

"When it comes to noticing just that the pauses are getting longer and longer, just measuring to the nearest round works for that; you don't need to measure to the nearest second.  Being able to notice the pauses are getting longer, is an argument for measuring the timing at all, because it causes some hypotheses to differentially lose 2s.  It's not an argument for measuring precisely."

"Among the remaining arguments, we have:  An argument that precise claims help by allowing right claims to lose fewer 2s, and wrong claims to lose more.  An argument that, if you don't measure precisely, you can't verify or falsify precise claims.  An argument about how precise measurements can rule out more theories, because if you can see closer than five hundred miles you can distinguish between claims about Egorian and claims about Ostenso."

"Those arguments are all valid, for having been spoken in Taldane.  But can anyone rephrase it more into the language of the Law of Probability, to fit it all into a common framework?  Tier-2s may now also answer."

"A more precise claim is one that - describes a smaller set of worlds," says Gregoria. "Not all the worlds where the person is in Cheliax, just the ones where they're in Ostenso, or just the ones where they're in this fortress. More precise claims are going to take more 2s to - find - and more observations are going to be evidence, about a more precise claim. Almost nothing I see is evidence about whether Asmodeus is 'great', because what does that even mean, but lots of things I see are evidence about whether Asmodeus expends more resources on the Material than other gods, and still more things I see are evidence - no, that's not precisely it, 'Asmodeus has thirteen tines in His crown' is a precise claim but almost nothing has bearing on it -"

" - okay, I think there is something there, I just need to nail it down. More precise claims point at smaller pieces of reality. It takes more twos to point to smaller pieces of reality. A theory that makes a more precise prediction is much more valuable, if it's right, it hit a smaller target. A theory that makes a more precise prediction is more easily wrong, which is just another way of saying what I just said - it couldn't be more valuable if it was right unless it was more easily wrong. You have to have a much better understanding of reality to make more precise predictions and have them still be correct. 

And your claim is only as precise as - if my measurement can't tell the difference between two claims, then I don't get any of the benefits of having made one rather than the other, a claim is only usefully precise if it's precise about what you're going to see, so, what your measurement is."

"I think possibly he's just looking for us to rephrase in terms of Problem 5 from that list of seven problems he gave us just before the artifact headband got dropped on Asmodia.  Which if I remember right was like - if you divide claims into a thousand parts and predict it down to the thousandth, that's like predicting that exact thing ten times as strongly than if you only say it's in a hundredth, but it could be in any of ten thousandths inside that and you're not saying which.  If you only measure it down to the hundredth, you'd - uh, I'm not sure what happens then, but you obviously can't be proven correct in the claim about thousandths."

"Yes, that's what I just said," Gregoria says.

Yes, that is EXACTLY what Alexandre was trying to say!

"Tier-1s."

"It doesn't take more 2s to point at smaller pieces of reality, it takes more 2s to point at smaller pieces of the probability distribution.  If you assign probability 1/64 to something, it takes you 6 'bits' to point there.  If you're already assigning 98/100 to some particular measurement coming in at exactly 0.891 seconds, you're only going to lose 0.03 2s each time.  Once you can exactly predict the measurements very certainly, they're not helping you narrow down things much further.  It would only automatically take more 'bits' to specify narrower pieces of reality if there was - some kind of - fixed probability distribution, or - this actually feels like it's pointing somewhere important but I don't know where yet."

"Precise claims don't have to talk about a smaller set of worlds, there can still be probability everywhere, it's that most of the probability will be concentrated in a narrower set of worlds."

"But theories only have so much probability to spread over all the possible precise measurements, so when there's more possible measurements, the probabilities on the vast majority of possible measurements have to be thinner.  Measuring things to three decimal places is one way to get lots of possible outcomes you're measuring over.  But it could also be something like - measuring three different things about it down to one-tenth apiece."

"If one theory puts lots of its probability within 0.002 seconds of 0.891 seconds, and another theory says 0.887 plus or minus 0.003 seconds, they've got some overlap, but measuring down to the nearest thousandth is pretty likely to do a good job of prying them apart.  Measuring down to the nearest hundredth instead, would be like adding up all the thousandths closest to that hundredth, to get the theory's predictions about what the measure would say as opposed to what was exactly real.  And then the two theories would give around the same probability to measuring 0.89, if you were only measuring down to the hundredth, and measuring at that precision wouldn't pry them apart much."

"You could maybe pry those two difficult theories apart more, even with only one hundredth precision available, if you constructed your experiment to measure them somewhere different."

"Theory 1 might say '7, 8, 9' takes 0.887s +/- 0.003s and Theory 2 might say '7, 8, 9' is 0.891s +/- 0.002s."

"But then you could also do an experiment at '107, 118, 129'. Maybe Theory 1 says that's 0.567s +/- 0.003s and Theory 2 says it's 0.128s +/- 0.002s. Your hundredth second resolution can detect that difference easily, it'll make one (or both) theories lose a lot of 2s in every measurement."

"You want to plan your experiments so they measure a place where your theories with the highest prior probabilities diverge more, so they're easier to distinguish from each other."