Pilar : "Seems like a good time to mention that I had that question about 'What's wrong with just testing a bunch of people and reporting whether or not there were at least five times as many all-14s who solved it?' And didn't ask because I am flawed and imperfect, and forgot the rules about what truly serves Asmodeus here."
Ione Sala: "I'm actually not sure how Keltham would answer that one? It's not thinking in probabilities, though. And what do you do if you test a few thousand people and 100 all-10s solve it and 499 all-14s solve it, report that the theory failed?"
Korva Tallandria: Korva is sitting on about ten different stupid questions at this point. She would just ask them out loud, at this point, if there was only one of them, but with ten it's hard to pick.
The most recent question is that she thinks she can see what Keltham's doing now - it's something about converting the results of the experiment into probabilities for different guesses about how the world is being the right guess, or something - but she has no idea why the calculations he did were the right ones in order to get those probabilities out the other side.
Ione Sala: Ione holds up a hand and seems to listen. "Okay, Message from somebody who says that their issue is actually that they've got no idea why the calculations you did on the whiteboard are how you convert the results of the experiment into probabilities for how those are the right guesses for how the world is."
Asmodia: Because alterCheliax students would not be this cautious and Asmodia is sick of it and Security is running Detect Thoughts anyways. This is Cheliax and they have ways of obtaining questions from reluctant askers.
Keltham: "...huh. Asmodia, you may have needed to do more basic Probability problems with your students, not just run them through all of the abstract high-level stuff, though I appreciate that you had sharply bounded time."
Asmodia: "Yeah, understood. I'll see if I can fix my fault there without taking up your own time."
Korva Tallandria: Korva didn't message, so instead of being nervous she's assuming that someone behind her also had one of her questions (not surprising, if Pilar also generated a different one of her questions independently), which makes her feel a little less bad about herself. Maybe she'll at least have another day to go over it and figure out what just happened.
Security: Amusing. Now the Security wants to see who works it out before they get their silvers after class, and the expressions on their faces if they don't.
Keltham: Meanwhile, the ever-oblivious Keltham is turning to gesture at his Prestidigitated whiteboard:
All-10s (2 YES, 3 NO):
Propensity: Likelihood:--------------- --------------1/10 (1/10)^2 * (9/10)^3 = 729/1000002/10 (2/10)^2 * (8/10)^3 = 1024/1000004/10 (4/10)^2 * (6/10)^3 = 3456/1000005/10 = 3125/1000006/10 (6/10)^2 * (4/10)^3 = 2304/1000008/10 = 512/1000009/10 = 81/100000
All-14s (3 YES, 2 NO):
= same but table flipped
"Right, so, in the first row, we've got the hypothesis, what if all-10s have a 0.1 'propensity' towards YES."
"Where, in this context, what that means, is:"
"Suppose the world is such a way that a random person from our 'survey-pool', about whom we know only that they have 10s in all mental stats of Intelligence, Wisdom, and Charisma, has a 10% 'propensity' to guess the correct answer on the 2-4-6 challenge within 30 minutes. Where 'propensity' is sort of a metaphysically fraught concept. But for our purposes in the kid's version of the lesson, 'suppose 10% YES propensity in all-10s' means, suppose the world itself is such that, from our perspective, it works out to 10% of the all-10s guessing within 30 minutes. Nethys, maybe, would know that some people in that pool were 99% likely to get it and some were 0.1% likely to get it, so it's not that the 10% chance of YES is an objective property of every single person. But from our perspective, the world is such a place that Nethys would say that around 10% of the people in that pool were 99% likely to get it, and we don't know which people."
"That's one hypothesis, the one in the first row. The second row's hypothesis is that all-10s in our survey-pool have a 20% propensity towards YES when we'll test them using our current experimental setup."
"Suppose we test five all-10 subjects - one after another, rather than in a simultaneous session - and see these exact results in order: NO, YES, YES, NO, NO."
"After getting a NO result with the first all-10 subject, we've seen something that was 90% likely in the world where all-10s have a 10% propensity to YES. But 80% likely in the world where all-10s have a 20% propensity to produce YES results."
"After seeing the YES from the second subject, we've now seen an additional fact that was 10% likely in the world where all-10s have a 10% YES propensity, and 20% likely in the world with 20% propensity."
"The two facts combined, that we've seen so far, are then 0.9 * 0.1 = 0.09 likely in the 0.1-propensity world, and 0.8*0.2 = 0.16 likely in the 0.2-propensity world."
Keltham: "We aren't saying anything there, one way or another, about the probability that the world is like that. We're not saying anything about the probability that Nethys would, if we could ask him or Ione could, write CORRECT on our sheet of paper, if we wrote down the guess, 'all-10s have a 10% propensity to YES'. To say that we'd need to know the 'prior-probability' of that hypothesis, and the 'prior' on all the other hypotheses, and have already computed the likelihood of all the other hypotheses, and 'renormalized' to get the 'posterior'."
"Which isn't a sort of thing that experimental reports try to do. Other people could know other evidence that would be relevant to whether Nethys was likely to write CORRECT on the guess. Stating probabilities like that are what prediction markets are for."
"What we're saying is - just flatly suppose that the world is in utter fact a place where all-10s have a 10% propensity to solve the 2-4-6 challenge within 30 minutes, relative to the experimental procedure we're using. Then it is valid, as a matter of 'logical-deduction', to say that you're 9% likely to get a NO followed by a YES on the first two subjects tested. It is likewise valid to say that, flatly assuming the hypothetical world where all-10s have a 20% YES propensity, you have a 16% chance of getting a NO followed by a YES."
"This is the key fact that other people need to know in order to update their beliefs based on your experimental results, so it's what the experimental report summarizes."
"It's a very local fact. It's like how, if you suppose that X=3 and Y=4, you can calculate that X*Y=12 without worrying about whether Z=5 or Z=7."
"If somebody else already ran tests on 100 subjects with all-10s, they might have seen results that pretty strongly updated them on the chance that Nethys would write 'CORRECT' on the 20%-propensity-hypothesis. They could've gotten 80 YES results, for example, which would make them pretty sure that hypothesis was wrong."
"But they can't have gotten any results relevant to the proposition that, in the hypothetical world where 20% of all-10s guess within 30 minutes given our experimental procedure, there's a 16% chance that we'll get a NO followed by a YES."
"You don't need to read all of the experimental reports in the world, you don't need to follow any prediction markets, to report that summary of the results you got."
"Suppose your third, fourth, and fifth results are YES, NO, NO."
"After the third result, we've seen data that's 0.9*0.1*0.1 = 0.009 likely in the 10%-propensity world, and that we'd have a 0.8*0.2*0.2 = 0.032 chance of getting in the 20%-propensity world."
"After the fourth result, a NO, we've seen things we're 0.0081 likely to get in the 10% world, and 0.0256 likely to get given 20% propensity."
"After the fifth result, a NO, it's 0.00729 or 729/100000, and... haha, whoops, 0.02048 or 2048/100000 for 20% propensity."
"This, by the way, being among the reasons not to trust your teacher even when he seems like such an uncomplicated straightforward reliable person, it is literally actually possible for him to be mistaken."
Keltham: Keltham Prestidigitates the corrected table of likelihoods:
All-10s (2 YES, 3 NO):
Propensity: Likelihood:--------------- --------------1/10 (1/10)^2 * (9/10)^3 = 729/1000002/10 (2/10)^2 * (8/10)^3 = 2048/1000004/10 (4/10)^2 * (6/10)^3 = 3456/1000005/10 = 3125/1000006/10 (6/10)^2 * (4/10)^3 = 2304/1000008/10 = 512/1000009/10 = 81/100000
All-14s (3 YES, 2 NO):
= same but table flipped
"I wish I had some way of asking you whether that helped, anonymous question-asker, but maybe communicate again to Ione if that was still unclear?"
Asmodia: Security, don't relay anything further about that topic to Ione unless somebody takes the initiative to ask. The meaning should be perfectly clear at this point, and if any newbies can't keep up and also can't ask questions, they're obviously headed for the dropout category.
Ione Sala: "Nothing yet, which hopefully indicates it was clear. And if not, well, they were adults and made their own decisions."*
(*) A saying now acquired from the kidnapped rescued Taldane girls in the secondary site. Though the same idea certainly does exist in Cheliax; indeed, there exist many Chelish variations on this saying, including several with Infernal loanwords in them and a couple straight from Hell.
Alexandre Esquerra: Alexandre raises his hand. "I have a question. If I have two hypotheses, say 40% and 60%, I know how to update them based on this data. But at all times -" Keltham has said, Alexandre obviously does not have it at all times "- you have a hypothesis that is - vague, not specific, 'what if all my guesses are wrong'. How do you weight the answer, 'anything but my hypotheses'?"
Keltham: "Good question! I'll ask you to hold that thought pending subtopic #3." Keltham gestures back to his previous list of subtopics he shouldn't forget to talk about. "Or actually I should maybe just write that one down..."
#1 - 'Published-experimental-reports' usually don't assign 'priors' or calculate 'posteriors', they just report all cheap details of the raw data, and maybe calculate some 'likelihoods' from obvious hypotheses
#2 - Separate experiments are usually supposed to avert 'conditional-dependencies', watch out for when that isn't true
#3 - If every obvious hypothesis has unexpectedly low 'likelihood' over all the combined data, it means the true theory wasn't in your starting set, often that different experiments had different hidden conditions
#4 - How to specially process the special meta-hypothesis 'all-other-hypotheses'
"Oh, and if that last Baseline word isn't translating, maybe the Taldane equivalent would be - everything we haven't thought of explicitly, all the theories we're not considering?"
"Anyways, this has hopefully ended up making #1 a little clearer."
"Going back to #2... what would make a good entrance point..."
"All right, so this isn't addressing #2 right away, just introducing an idea we'll use there, but."
"Suppose that we tried summarizing the hypotheses here into three buckets. One bucket that the YES-propensity is 0.1 or 0.2 or 0.4, one bucket that the propensity is 0.5, one bucket that the propensity is 0.6 or 0.8 or 0.9."
"Is it then possible to describe the likelihood of our data NO YES YES NO NO, conditional on the first or third bucket? For the middle bucket it's obviously 3125/100000 or 1/32."
Asmodia: ...why did the only candidate who's letting herself generate responses have to be her.
Yes. But BE FUCKING SHY OKAY.
Willa Shilira: She makes sure to look back and forth before answering, as if she's making sure nobody else is going to say anything.
"It'd have to be the sum of the likelihoods right, or the whole idea of probability wouldn't make any sense? So 0.1 or 0.2 or 0.4 would be total likelihood of (729+2048+3456)/100,000 = 6233/100,000 vs 0.5 at 3125/100,000"
"Except that only works if you thought 0.1 and 0.2 and 0.4 had the same chance to begin with, before you did the experiment. And even 0.5 too, as likely as all the others. If you thought they all had different before-chances it would mess up the weights and I'm not sure doing that would work anymore. Maybe you'd have to do them individually if you couldn't trust they started off the same?"
Keltham: "Suppose I've got a coin that might have a 4/10 chance of coming up Queen, or maybe a 6/10 chance of coming up Queen, I'm not really sure, seems about equally likely to be each. Would you agree that in this case, the coin is (4+6)/10 = 10/10 likely to come up Queen?"
Willa Shilira: Oh no something's definitely wrong, but she can fix it, she knows she can fix it it's just math...
"No, definitely not, but this is relative likelihoods and not absolute likelihoods? Maybe that doesn't help, hmm"
"Is the idea that we have to divide by the number of cases our bucket used for it to work? And maybe that the before-chance we use then is the sum of the before-chances of all the pieces of the bucket together?"
Keltham: "Stop trying to think so much in complicated math and fall back on common sense. I spin one fair coin, 50% chance of Queen, 50% of Text. If the first coin comes up Queen, I take out a new coin that's biased to have a 60% chance of coming up Queen. If the first coin comes up Text, I take out a different new coin that's biased to have a 40% chance of coming up Queen."
"Suppose I was actually going to carry out that procedure. You don't get to observe the prior coinspin. What's your betting chance that, when I spin whichever new coin I take out, it shows up Queen?"
Willa Shilira: "It has to be 50%, which is the same as 0.5*0.6 + 0.5*0.4. But if the first coin is in this case the before-chance, we don't know that it's 50/50, do we?"
She has ever gone through cycles like this getting tortured a few times, but being tortured is what happens to good Asmodeans when they're wrong, right?
Keltham: "I mean, in this case, you know because I told you to assume that setup, and answer me conditional on the assumption."
Alexandre Esquerra: "To answer the question," drawls Alexandre, looking up from his scribbled notes, "if we assume that 0.1, 0.2, and 0.4 all start with equal prior likelihoods, and similarly for 0.6, 0.8, and 0.9, it's (729/100000 + 2048/100000 + 3456/100000), divided by three because there are three equal possibilities, compared to (2304/100000 + 512/100000 + 81/100000) divided by three for the same reason, gives you -" (he's been doing calculations while Willa talks) "- roughly two thousand seventy over one hundred thousand, as opposed to a little under seven hundred over one hundred thousand, giving you a ratio of a little under three to one in favor of the first bucket of theories against the second bucket."
Wait no that's obviously wrong oh damn it -
Korva Tallandria: "Does that work?" asks Korva, sounding genuinely unsure. "That gives you a lower probability for both of the buckets that had three hypotheses in them than for the bucket that had only one hypothesis, even though the results seem more consistent with a true answer of .4 than with .5."
Keltham: "That's correct! If you spin a fair three-sided coin and then, depending on the results, start flipping a new biased coin with 0.1 or 0.2 or 0.4 propensity to Queen, your chances of getting Text Queen Queen Text Text are around 2%. Whereas if you just start spinning a fair two-sided coin, your chance of getting Text Queen Queen Text Text is around 3%."
"The 0.4 sub-hypothesis is most likely to generate Text Queen Queen Text Text. But that hypothesis starts out with only one-third of the prior probability mass in the 'less than 50%' bucket of hypotheses. The bucket as a whole is less likely to generate the sequence than the sub-hypothesis inside it."
"Incidentally, one would not in Baseline say that the sub-hypothesis of 0.4 propensity started out with a 1/3 'likelihood' of being true, but that it started out with a 1/3 'prior-probability' of being true. Some of the reason why I've been using those Baseline words is because they have precise meanings, whereas Taldane has a bunch of words like chance and probability and likelihood that all apparently mean the same thing why does this crazy language even have that don't actually answer me unless it's important somehow."
Alexandre Esquerra: - Wait, Keltham thinks he's right??? But he's wrong!
...Alexandre thinks he's just going to stick with 'the odds of him failing the most important class in his life have gone down', and bedamned to the truth.
Korva Tallandria: She's so confused???
...she's going to raise her hand. Because she feels like she's being confused in a smart way and not a dumb way, for once, but if they've belabored this point for too long then he can ignore her. That's how raising hands works, right?
Keltham: Points finger at Korva.
Korva Tallandria: "I think I must be confused about how the buckets work. Because, like - say we have one hypothesis that the true value is exactly 50%, and one hypothesis that the true value is less than 50%, and one hypothesis that the true value is more than 50%. It seems like the likelihood - or probability, sorry, I didn't catch the distinction - it seems like it must be more likely that the true value is below 50% than that it's exactly 50%, even though the below-50% space also includes values like 0%, which has obviously already been outright disproven. So - I guess I feel like if .4 is more likely to be right than .5, I don't see why the hypothesis that covers a bigger space, that also includes the value that the true value is most likely to be closest to, becomes less likely just because that hypothesis also includes values that are much less likely to be right than some other narrower hypothesis. - I'm not sure I said all that right."
Keltham: "Okay, wow, people's intuitions about probability work really differently when they haven't been raised as dath ilani children. I think probably you just need to invent a whole lot of problems and play around with them, the way anyone - well, the way any dath ilani kids would do as kids?"
"Let's say I spin a fair twosided meta-meta-coin... let all coins be assumed two-sided and assumed fair unless explicitly stated otherwise."
"Anyways, new procedure. First, I spin a meta-meta-coin. If the meta-meta-coin comes up Text, I spin an 'objectlevel'-coin five times. If the meta-meta-coin comes up Queen, I spin a three-sided meta-coin. Then depending on the result of that coin, I fivetimes spin a biased 'objectlevel'-coin with 0.1 or 0.2 or 0.4 propensity to produce Queens."
Keltham will attempt to whiteboard this:
"It's actually just true that if you don't know any of the meta-spins, and just see my unknown objectlevel-coin producing Text Queen Queen Text Text, there's roughly two chances in five that my meta-meta-coin came up Queen and picked a biased coin, and three chances in five that my meta-meta-coin came up Text and picked a fair coin."
"Why? Because when you end up with a biased coin, it's sometimes biased 0.4, but two-thirds of the time biased 0.1 or 0.2. Mostly you'll see fewer Queens, when the meta-meta-coin comes up Queen. When you see Text Queen Queen Text Text, that could be because the meta-coin was flipped and selected the 0.4 coin, but more likely, the meta-meta-coin selected a fair objectlevel-coin and that fair objectlevel-coin happened to produce two Queens and three Texts."
"A dath ilani kid would now be 'programming' a 'computer' to run a million simulations of this procedure and show them how many cases of Text Queen Queen Text Text were generated by the meta-meta-coin coming up Queens versus Text and verifying that's how it actually played out. Here... we'd need to find a three-sided coin and a ten-sided coin, or maybe a cube and an 'icosahedron', regular 20-sided solid, to spin. And then we'd probably have to do a few hundred spins to collect enough 'data' for the 'statistics', if the math didn't feel intuitive."
"But I'd hope that - the first time the meta-meta-coin came up Queen, and you spun a meta-coin and it selected 0.1, and the resulting objectlevel-coin produced Text Text Text Text Text, it might become more intuitive why, when the meta-meta-coin comes up Queen, what you mostly expect to see is mostly Text. So when you don't see that, it's not likely the meta-meta-coin came up Queen."
Korva Tallandria: "That... makes sense, but how do we know that there isn't instead a four-sided meta-coin that picks among all the possible coins, including the fair one?"
Keltham: "In terms of this particular problem? Because I told you so."
"Why did I tell you so? Because I was trying to pump the intuition that - assuming 0.1, 0.2, 0.4 equally prior-probable within the 'less than 0.5' bucket - that bucket was then less 'likely' to yield Text Queen Queen Text Text than the 0.5 bucket, even though 0.4 was in that bucket. I was trying to pump that intuition by showing that, if the whole biased bucket and the whole unbiased bucket started out equally probable, then, after seeing Text Queen Queen Text Text, we'd think the unbiased bucket had become more probable and the biased bucket less probable. So what we saw must have a lower 'likelihood' in the biased bucket that starts out with 0.1, 0.2, and 0.4 having equal prior-probabilities."
"I mean, we could argue about how that would go in real life, instead of a thought experiment. You could say that all four hypotheses are equally simple to describe out loud and should therefore be around equally probable. I could then counterargue that if we're talking about an actual coin, then in real life, most coins are probably pretty close to being fair random-number generators when spun - though I ought to actually verify that here, before I bet anything important on it. So it should actually take hundreds of observations before we start believing the coin is 40% biased towards Queen, I would argue; five coinflips is nowhere near enough. Therefore, I'd conclude, 'the coin is biased 40% Queen' is a lot less likely than 'the coin is an unbiased random generator'."
"But it would be better if arguments like that didn't have to appear in our 'published-experimental-reports'. Which is one angle towards 'grokking' an underlying central reason why 'published-experimental-reports' ought to summarize likelihoods for hypotheses that are more like 'observational likelihood if this coin has 0.4 Queen propensity', and less like 'observational likelihood if this coin has a less than 50% Queen propensity'."
"If you just summarize for the reader 'What is the likelihood of my data, in the world where the coin comes up 40% Queen? The world of 50% Queen? The world of 10% Queen?' then you don't have to confront the question of whether 40% Queen was 1/3 as prior-probable or equally prior-probable with 50% Queen."
"Oh, and, uh, to make it explicit:"
Examples of Baseline terms for 'prior', 'likelihood', 'posterior':
'Prior' coin is 40%-Queens:
P( Q=0.4 ) = 1/6'Prior' coin is 50%-Queens:
P( Q=0.5 ) = 1/2'Likelihood' of TQQTT, if coin is 40%-Queens:
P( TQQTT ◁ Q=0.4 ) = 0.03456'Likelihood' of TQQTT, if coin is 50%-Queen (fair):
P( TQQTT ◁ Q=0.5 ) = 0.03125'Posterior' coin is 40%-Queens, after seeing TQQTT:
P( Q=0.4 ◁ TQQTT ) = 3456 / (729 + 2048 + 3456 + 3125*3) >(1/5), <(1/4)'Prior' coin is <50%-Queens:
P( Q<0.5 ) = P( Q=0.1 ) + P( Q=0.2 ) + P( Q=0.4 ) = 1/6 + 1/6 + 1/6 = 1/2'Likelihood' of TQQTT if <50% Q:
P( TQQTT ◁ Q<0.5 ) = 1/3 * .00729 + 1/3 * .02048 + 1/3 * .03456 ~ 0.02
"How am I doing, Korva Tallandria?"
Korva Tallandria: "I - think I get that, as far as it goes, with the coins? But I don't immediately see how it applies to the people. - if it applies to the people the same way, I'm assuming it does but I'm not sure I should be."
Keltham: "I mean, the sense in which it also applies to the people, is that your report should summarize the 'likelihood' that your results were generated by a 10% propensity for all-10s to guess within 30 minutes, not the 'likelihood' that your results were generated by a less than 50% propensity for all-10s to guess within 30 minutes. Because to do the latter thing you have to make a bunch of weird assumptions, and your math is just going to get more and more needlessly complicated as you dig yourself in further."
"And by way of showing how much further into complicated trouble you'd end up digging yourself:"
"Again, let's say we were going by bucketed hypotheses. One hypothesis, the meta-coin hypothesis, says that there's a 1/3 chance we live in a world where all-10s have a 10% propensity to solve 2-4-6 in 30 minutes, 1/3 chance it's 20% propensity, 1/3 40%. The other hypothesis, the fair-coin hypothesis, says we live in a world where all-10s have a 50% propensity to solve in 30."
"We don't actually need to consider the probability of these two hypotheses relative to each other. Let's say we test five all-10 subjects and get NO YES YES NO NO, meaning two subjects guessed within the time limit, three didn't. Our experimental report is just going to summarize the 0.2 'likelihood' of the data assuming the meta-coin hypothesis bucket, and the 0.3 'likelihood' of the data assuming the fair-coin hypothesis. That's true regardless of the 'relative prior-odds' of the two hypotheses relative to each other."
"So we publish our report. 0.2 likelihood for the less-than-50% bucket, 0.3 likelihood for the 50%-propensity hypothesis. There's a questionable assumption that 10%, 20%, and 40% were all 1/3 likely assuming the propensity was under 50%, but fine, whatever, we've got to assume some 'prior distribution' to report a combined likelihood on that whole bucket all at once, yo."
"Along come some replicators. They test 5 more people. They get YES NO NO NO YES, so also two subjects who guessed and three who didn't."
"Now what? What does the combined evidence say? Anybody want to give the obvious wrong answer?"
Ione Sala: "Obvious wrong answer: This new data also has 0.2 likelihood on the less-than-50% bucket, and 0.3 likelihood on the 50% bucket, so the combined likelihood across the two experiments is 0.2 times 0.2 equals 0.4... equals 0.04 assuming less-than-50%, and 0.3 * 0.3 = 0.09 assuming 50%."
Keltham: "And why isn't that just totally right? Or maybe I'm trolling you by calling it the wrong answer, and it is right? Candidates only, you can message Ione if you're worried your reply is stupid."
Aevylmar: "Are we still not considering any other - hypotheses, and just interested in the relative ratios?"
Keltham: "Yup!"
Aevylmar: "We've become more confident, which sounds right... does this get the same result as if we did one experiment with ten people in the first place? I'm not sure if it does..." She's going to start checking the math.
Keltham: "Well, everyone else do feel free to follow along and try the math on that part. Raise your hand when done, practice accuracy before you practice speed."
"After all, a Lawful way of looking at the world shouldn't care whether you call your collected 'data' by the name of one experiment or two experiments. You should always get exactly the same answer either way. Though, to be sure, you might have occasional occasion to notice that different 'data-subsets' might have been collected under possibly different conditions."
Pilar : Pilar raises her hand before anyone else. She's faster than even Asmodia once she knows exactly which rules to follow and which procedure to execute, and in this case she knows exactly what to do.
Aevylmar: The first of the new researchers to finish speaks up immediately - "About Fifty-five hundred and sixty-seven to ninety-seven hundred sixty-six," she says, "which really isn't the same thing as four to nine."
Keltham: "Um, sorry for not being explicit - if I say for people to raise they're hand when they're done, that's so I know everyone is done, and meanwhile, everybody gets a chance to try on their own before hearing anybody else's result."
"At least you didn't say how you did the calculation, so others can, perhaps, do their own calculations and see if they think yours is correct."
Keltham comes over to look at the scratch paper, if any; what calculation seems to have been done?
Aevylmar: Squared both sides of 729/100000, 2048/100000, and 3456/100000, divided it by three, then compared it to 3125/100000 with both sides squared, then canceled out the denominator since it was the same on both sides of the equation, with some approximations for doing faster arithmetic.
Keltham: Message to her: Don't smile or anything, but: Correct. Though two significant digits would've been fine there.
...and she was first of the newcomers, so probably everyone is trying to compute needlessly many digits, which, okay, fine.
Keltham: Keltham will wait until everyone's raised their hand, and then go to the whiteboard and show how he'd have approximated it:
0. 1/3 * 0.00729^2 + 1/3 * 0.02048^2 + 1/3 * 0.03456 ^ 2 vs. 0.03125^21. ~ (2^2 + 3.5^2)/3 vs. 3.1^22. ~ (4 + 12.25) / 3 vs. 9.613. ~ 5.4 vs. 9.6 (/6 * 10 => proportional to 9 vs. 16)4. ~ .00054 vs. .00096
"...where step 1 is dropping the first term that's obviously going to end up insignificant, multiplying both sides of everything by 100, and rounding to two significant digits. I mean, we wouldn't do that in Civilization because we have 'computers' to do the calculation for us, and here you might not do it in a Science! report, but it's definitely fine in my classes."
"And step 4 is dividing again by 10,000 to put that factor back in, because your experimental report should summarize the absolute likelihood of the data, not just the relative likelihood of the data. Somebody who wants to compare some completely different hypothesis's likelihood, one you didn't even consider, needs the absolute likelihoods to do that - they need the .00054 and .00096 version, not the 9 vs. 16 version."
Keltham: "Imagine that we've got two separate research groups both testing the hypothesis that all-10s solve 2-4-6 with 50% propensity, or alternatively, less than 50% propensity. They each don't know the other group exists; however, they both use the - hypothetically for this thought experiment - universally standard rule that 'less than 50% propensity' is of course best-modeled in practice by three equally weighted 'probability point-masses' on 0.1, 0.2, and 0.4."
"The first group reports a likelihood of 0.02 on the less-than-50% metahypothesis, and a likelihood of 0.031 on the 50% hypothesis."
"The second group reports the same thing."
"The way we've set up the hypotheses being reported on, we cannot just multiply the two likelihoods together. The task of combining evidence from different 'published-experimental-reports' is now a big complicated deal requiring us to recheck their original data and redo all their calculations."
"Alternatively, if they had both reported likelihoods of 0.007, 0.02, 0.035, and 0.031, on the distinct hypotheses of 10%, 20%, 40%, and 50% propensity respectively, we could have just multiplied the likelihoods together from both groups, and our ability to accumulate data from across multiple experiments would be vastly simplified."
"Of which it is said out of dath ilan, to those dath ilani children who need to hear it: Different 'effect-sizes' are different hypotheses."
"That, Carissa, Pilar, is why we can't just have the hypothesis that all-14s have at least five times the propensity of all-10s to solve 2-4-6 in 30 minutes. We can look at the data and see if that actually happened or not. But as soon as we try to figure out the exact likelihood that it happened, we are cast into a nightmarish multiverse of different ways the world could be, such that the statement 'all-14s are more than five times as likely to solve in thirty as all-10s' is true about worlds like that, all of which have different likelihoods of yielding the data we saw."
"Like, just on this breakdown, that could be because the chances were .2 and 1.0, or .1 and .5, or .1 and .6, or .1 and .8. And every one of those hypothetical propensity '2-tuples' will yield a different likelihood for whatever data we saw. So you'd have to put a prior on their relative odds inside that metahypothesis bucket, before you could calculate the likelihood for the whole bucket."
"And then, actually seeing any data, would update the odds inside that bucket, which would change the likelihood for any future experiments, even if the replicators saw exactly the same data you did."
Keltham: "And that takes us to the principle I wrote earlier on my todo list:"
#2 - Separate experiments are usually supposed to avert 'conditional-dependencies', watch out for when that isn't true
"What I mean here is that - when you are otherwise doing things correctly - it should usually be the case that, for the likelihoods that the 'published-experimental-report' is summarizing for different hypotheses, if some replicators came along and did their own experiment, their likelihoods should be something they can calculate independently of your data. It shouldn't be the case that they have to look at your data, to figure out the likelihoods given their hypotheses."
"This in fact is the property that lets us compute the joint likelihood of a hypothesis across two experiments, by multiplying the likelihoods together from the separate experiments. Symbolically:"
P( data from first and second experiments ◁ the hypothesis )= P( data from 1st experiment ◁ the hypothesis ) * P( data from 2nd experiment ◁ the hypothesis) if and only ifP ( data from 2nd experiment ◁ the hypothesis ) = P( data from 2nd experiment ◁ the hypothesis & data from 1st experiment)
"When you say, 'maybe all-14s have 60% propensity to solve in time, and all-10s have 10% propensity to solve in time', you're describing a way reality can be, where the likelihood of my found pattern of YES and NO responses, if that's true, is just the same no matter what data you found in your own experiment. Maybe your data made that world look incredibly improbable, but that doesn't matter; I can still answer the question of how likely my data would be, if that world was the case, without looking at your data."
"When you say, 'maybe all-14s have at least five times the propensity of all-10s to solve 2-4-6 in 30 minutes', that is a way the world can be; but it's a way the world can be, where calculating the likelihood of my data in that world, requires me to make up a bunch of prior-probabilities, and then those probabilities change depending on the data that you got."
"Which makes it immensely complicated to quickly look over the summaries of what different people's experiments said about different worlds, and combine them together into a joint summary of what reality has told us about them all."
"It would, in fact, be possible to combine a lot of little experiments all of which suggested that - if you wrote the summary this way - the data was more likely if all-10s had 50% propensity to solve, versus less-than-50% propensity to solve, and get out a new update that the combined data was more likely if all-10s had less-than-50% to solve. If you multiplied enough 0.035 likelihoods from the 40%-propensity hypothesis, compared to the 0.031 likelihoods from the 50%-propensity hypothesis, then eventually the 40%-propensity hypothesis would come to dominate the predictions of its bucket, and then its bucket would start to dominate the other hypothesis."
"Which paradoxical-seeming combination, again, doesn't happen if you consider the 40%-propensity hypothesis separately, because then it's clear from the start that 40% propensity is gaining on 50% propensity in each experiment."
"Hence again the proverb: Different effect sizes are different hypotheses. Which argues again against thinking that 'all-14s are at least five times as likely to solve as all-10s' is a good way to split up the world for purposes of SCIENCE! Even though, in terms of 'truth-functional' scaffolds, it is a way the world can be. It could even be the metafact that is useful and that we're interested in. We should still ask the Science! question first, what are the exact real effectsizes, and then check the useful metafact afterwards."
Keltham: "Likewise if you started thinking that 'this coin isn't random' or 'this coin is biased to favor Queen' was a good way to describe the hypothesis you were considering. If two experiments show that the same coin is probably biased to Queen by notably different amounts, they're pointing at incompatible ways the world can be, and something is wrong, some condition has changed between experiments, at least one group is screwing up."
"You definitely wouldn't say, 'Well, our hypothesis was that this coin was biased to favor Queen, and group one spun it a bunch of times and found that it came up Queen 900 times out of 1000, and group two spun it a bunch of times and it came up Queen 520 times out of 1000, and both of those results are instances of 'the coin came up Queen more often than it came up Text', so both confirm the hypothesis that 'the coin is biased Queen', and the experiment has 'reproduced'. You are actually less confident after two apparent confirmations of your original statement than you were after one confirmation, because in the details of the particular worlds, it's clear that something was wrong with at least one experimental setup."
"But that apparent paradox is just an artifact of bucketing together different ways the world could be, that yield very different likelihoods on the exact data observed, into one metahypothesis of 'this coin is biased to favor Queen'. If you said instead 'the coin yields 90% Queens' or 'the coin yields 52% Queens', there would be no illusion of the first experimental result agreeing with the second result, there would be no illusion that the result had 'reproduced'. Fix a local hypothesis, a single effect-size in this case, that makes the data have independent likelihoods between one experiment and another, that fully specifies the likelihood of the data as a matter of logic, and doesn't change when we read other experimental reports. Summarize the likelihoods for hypotheses like that, and it would be clear that the data from one experiment was compatible with an exact hypothesis, and the data from the other experiment was not."
"Which, uh, yeah, the lesson is, there are these careful precise details about how to do SCIENCE! correctly, and those details actually matter a lot for making your whole Civilization's SCIENCE! output fit together and have the whole thing make any sense. Even for a small project like ours, it's still probably best to do it that way, if we want things to make any sense."
Korva Tallandria: Korva, who was the last to raise her hand for the last exercise, and who has been calculating through another panic attack for the past twenty minutes, has now realized that the horrible feeling of wasting lots of time making errors that she still doesn't understand on stupidly complicated problems was the point of all of this, that this incredibly painful classroom experience has all been an illustration specifically for the benefit of Carissa Sevar about why they shouldn't do things the way she suggested, even though there is no outward indication that the Chosen of Asmodeus herself experienced any distress about it at all at any point.
Korva thinks that, probably, if she were in charge, she would have experimenters report their data, and then everyone who wanted to see how well that data fit some specific hypothesis could CALCULATE IT THEIR OWN GODSDAMNED FUCKING SELVES.
dath ilan: There are, even in dath ilan, children who will agreeably acquiesce in doing SCIENCE! the same way everybody else does it, without carefully detailed demonstrations of exactly what grimdark dooms will befall them if they try to make up their own methodologies that violate the coherence constraints.
Tiny Keltham was not one of those children.
And besides, even if you report the raw data like a sane person, people do need to know which calculations to do after they've got all the data, right? There are some local calculations that do tesselate together neatly to global calculations, and you might as well summarize those when they exist. Which requires you to know how to set up the kind of calculation that modularizes well, and distinguish it from calculations that don't.
Willa Shilira: Willa feels like she's following well, but she did think she was before, and then she still managed to get herself tied up somehow. In front of everyone, as usual.
Though in retrospect she thinks she had it but there was just a lot of confusion about before-chances priors and some miscommunication? But overall she's happy that this particular weird mess of logic is going to stay away from SCIENCE! The bad buckets were hurting it.
She doesn't know SCIENCE(!) well enough to feel protective of it yet, but she expects she will. Except that expecting yourself to be convinced of something in the future doesn't make any sense. So she might as well feel protective of it already.
Alexandre Esquerra: Alexandre is apparently quite good at probability theory, since he was following for all of that! He is therefore happy that his knowledge (a) is growing and (b) will probably continue to grow, since he's not permanently behind everyone and can therefore continue taking the MOST IMPORTANT CLASS IN THE WORLD so he can obtain MASTERY OF REALITY. Alexandre's power and knowledge continue to grow, all hail Lord Asmodeus.
Keltham: "Also, just to be clear, while you wouldn't necessarily get paid very much for it except in very special cases, there is such a thing as just running out and trying a bunch of stuff that you didn't preregister and aren't going to be especially careful about analyzing if you even report the results at all. That's definitely still 'science', the individual kind, even if it's not 'SCIENCE!' the project of all Civilization. It is not considered foolish to do noncareful nonrigorous experiments to figure out which careful rigorous experiments to do, before you spend a lot of effort on writing up something to preregister."
"That's, uh, basically what I did the previous day when I was experimenting on boiling acid and having it explode a lot. I wasn't exactly trying to lay a firm foundation for future Golarion Science!, just poking around rapidly and trying to get a handle on how 'chemistry' worked with magic in the mix."
"This, too, is something of a dath ilani stereotype. Stereotypically, it sometimes involves an Ione-like person who tries to prevent the maniacal-scientist from accidentally killing himself. They are even, often, a girl and a boy respectively, and it is not unknown for them to end up romantically involved. Though I have to ask again, do I really need one of those when I already have several girlfriends and my new universe comes neatly equipped with healing and resurrection spells?"
Ione Sala: "YES because resurrections are EXPENSIVE and they are MUCH MORE EXPENSIVE when the person has COMPLETELY DISSOLVED THEMSELVES IN ACID."
Keltham: "Oh. Huh. I wasn't actually aware that it got more expensive if I completely dissolved myself. Good thing to watch out for, then."
Ione Sala: Why is her life like this.
Ione Sala: Even Nethys can't have been this bad. He wouldn't have survived to become a god.
Nethys: ...there may possibly have been several romantic interests involved in his continued mortal existence.
Carissa Sevar: "I"m actually a little surprised there's not more specialization. If I were imagining how to do this I'd have some people write detailed experiment-specifications based on what they wanted to learn, other people follow the experiment-specifications, and then the first people or other specialists do analysis of the results to figure out how to bet."
Keltham: "It's specialized out to the Fourthplanet colony and back again. There's going to be literally hundreds of people in Civilization who do nothing but talk to the big chemistry outfits to figure out exactly what they need and write up exact experimental specs for experiments involving 'sulfuric acid' in particular."
"Thing is, everybody in that process needs to understand the rules on how the greater Science! process works and why. You can't write up a good experimental spec unless you know how a replicator will carry it out and what an analyst will do with the data. You can't be a good analyst unless you know what was really going on in the experiment, what the impact-buyers were originally asking for, and what the report-writers need from you."
"Good specialists don't have to be good generalists, they can be mediocre generalists, but they do have to be generalists. You don't need to be able to write a pro-level saleable experimental-report before you can hire a writer to do that for you. You do need to be able to write a terrible, unreadable report that is nonetheless judged by a professional skill-evaluator to contain all of the necessary details."
"There's literally nobody in the process who doesn't need to understand the Law of Probability well enough to know the difference between hypotheses-that-fully-imply-likelihoods and semantically-well-formed-propositions-that-underspecify-likelihoods and word-sequences-that-don't-even-have-clear-truth-functionals-yet."
Keltham: "Remember, I am not in fact, by Civilization's standards, a great experimentalist, a famous analyst, the one who brings the light of cause-selection to the world."
"I'm a teenager what got tossed into another universe, is what I am."
"Everything I'm teaching you is what Civilization makes sure to teach all of the eight-year-olds, so they'll be able to read the newspapers when they grow up."
"I mean, not literally. Not literally actually. Not every eight-year-old has read a couple of novels in which the protagonist has to improvise scalable production of sulfuric acid, in which, unfortunately, most of the memorable drama was about safety precautions obviated for us by Resist Energy (Acid). I know more than the bare basics, in a few places, because I thought they seemed cool to know."
"But definitely everything we've been covering today has been like that."
"There's a saying out of dath ilan, 'Professional specialization is what grownups do when they actually want results.' But don't go setting your eyes on a specialist's glory just yet. Let's all get to the eight-year-old generalist level first."
"If, like... we can successfully do even that, without me having actually been a professional specialist in teaching eight-year-old Law of Probability."
Keltham: Keltham's pocketwatch shows he's running out of time on his third and final Communal Share Language (Baseline), which provides something of a helpful natural time limit on the morning's lecture, so he'll now try to race through some of his remaining whiteboarded pending-topics.
Keltham:#3 - If every obvious hypothesis has unexpectedly low 'likelihood' over all the combined data, it means the true theory wasn't in your starting set, often that different experiments had different hidden conditions
Let's say you're testing (only) the hypotheses for 20% propensity-to-solve and 60% propensity-to-solve.
Suppose you test 300 subjects, and 100 of them solve it.
At this point, obviously, they're not going to be multiplying things out by hand any more, and are going to be working with logarithms...
log2 (0.2) = -2.322log2 (0.4) = -1.322log2 (0.6) = -.737log2 (0.8) = -.322
(bonus points, Keltham declares, for anybody who already noticed that 2 entries in this table are blatantly redundant, but he's writing them out anyways for ease of reference)
2s lost, 20% propensity:-2.32 * 100 + -.32 * 200 = -2962s lost, 60% propensity:-.74 * 100 + -1.33 * 200 = -338
Now, if you are a naive six-year-old being led astray by an older child trolling you, you might look at this, and proclaim that the 20%-propensity hypothesis did much better than the 60%-propensity hypothesis, by a likelihood factor of over a trillion.
But really what this data is yelling at you is "The true hypothesis was not in your explicit hypothesis set!"
Why? Consider about how well the two theories say they ought to do, loss-of-2s-wise, in the worlds where they are actually true:
Expected 2s lost given 20% propensity:-2.32 * 60 + -.32 * 240 ~ -2&1/3 * 60 + -1/3 * 240 = -140 + -80 = -220Expected 2s lost given 60% propensity:-.74 * 180 + -1.32 * 120 ~ -3/4 * 180 + -1&1/3 * 120 = -135 + -160 = -295
...which is to say that both hypotheses lost way more than they expected to lose.
This is a sort of hint that tells you to look for a new hypothesis, like "1/3 propensity", say. It holds even if you get the subtler hint that some hypothesis is doing way better than it expected to do, losing fewer twos than it said it should lose.
There's obviously ways to think about all the possible propensities at once, but Keltham doesn't think he can get to those this morning, given that they don't have calculus yet.
Keltham: There's a similar phenomenon that would suggest two experiments were working under different conditions, even if the data got all mixed together before you looked at it. Say you were gathering data to find out the average Intelligence in Cheliax, which you expect to be the sum of lots of tiny factors and hence distributed along a central distribution.
Actually, however, your data was gathered for you by a professional data-gathering firm, though, uh, you might possibly have not done a lot of due diligence before hiring them. They were cheap, though!
This data-gathering firm immediately subcontracted out your job to two even cheaper subcontractors.
What these data-gatherers should have found - at least if the data told to Keltham himself was correct, and a couple of other facts seem to have borne it out - was that Golarion has a mean Intelligence of 10, with a square-root-of-average-squared-deviation-from-the-mean of 2. (Baseline: 'Deviation' of 2.)
One subcontractor, however, didn't spell-check their survey, and the spelling errors turned off the smarter and more perfectionist people reading it, so their biased sample of respondents had average Intelligence 8 and deviation 1.
Another subcontractor went where it was very convenient for them to find survey respondents, which was, it turned out, people standing in line to apply to a wizard academy. That subgroup had average Intelligence 12 and deviation 1.
If both datasets are completely mixed together before you get them, when you compute the average, you'll find it's around Intelligence 10, and the deviation... will not be exactly 2, but it will be around 2.
But the hypothesis "This is a central distribution with average 10 and deviation 2" would predict that 10 is the most likely Intelligence score you can find. Intelligence-10s will actually be relatively rare if your distribution is the sum of two subdistributions with deviation 1 and averages 8 and 12. 6% or so of subjects will have Intelligence 10, instead of 38% as the hypothesis predicted. You don't need to notice that particular deficiency by looking at Intelligence-10 subjects specifically. It'll show up in the combined likelihood of all the data being much lower than expected, even if the whole thing is calculated by a 'computer' that wasn't 'programmed' to detect that exact kind of anomaly.
You can calculate what kind of score you'd expect to get, if any of your hypotheses were actually true, and if all the hypotheses score much lower than they expect, they're all - in Baseline colloquialism - 'stupid with respect to the data'. This doesn't always happen when different experimenters are working under secretly different conditions and measuring actually different phenomena, it is not always obvious just from the likelihoods especially if you mix all the data together before checking it, but it is an example of a pattern suggesting that the true hypothesis wasn't anything you were considering.
One should always keep in mind, though, that the 'fair coin' hypothesis never looks stupid no matter how much pattern it's missing out on. If you spin a coin 1000 times, and it comes up Queen 1000 times, the fair-coin hypothesis expected to lose 1000 twos and that's exactly what it loses. In a case like that, you have to think of the specific better hypothesis - 'this coin has a 100% propensity to Queen' - or perform some more general test that implicitly takes into account the possibility of lower-'entropy' hypotheses like that - before you can see the problem.
If it's just never occurred to you that coins might be biased, if you haven't invented any tests to detect biases, then contemplating the fair-coin hypothesis alone is not going to show you that hypothesis doing any more poorly than it promised you it would do.
Keltham:#4 - How to specially process the special meta-hypothesis 'all-other-hypotheses'
Okay so according to his pocketwatch Keltham has two minutes left to tackle this one before Share Language runs out, and that is not really a lot of time for what is actually the deepest question they've come across so far.
There are always better hypotheses than the hypotheses you're using. Even if you could exactly predict the YES and NO outcomes, can you exactly predict timing? Facial expressions?
The space of possible hypotheses is infinite. The human brain is bounded, and can only consider very few possible hypotheses at a time. Infinity into finity does not go.
The thing about all the possible hypotheses you're not considering, though, is that you are not, in fact, considering them. So even if - in some sense - they ought to occupy almost-1.0 of your probability mass, what good does it know to do that? What advice does it give you for selecting actions?
And yet there is advice you can derive, if you go sufficiently meta. You could run that test to see if all of your hypotheses are scoring lower than they promised to score, for example. That test is not motivated by any particular hypothesis you already did calculations for. It is motivated by your belief, in full generality, in 'the set of all hypotheses I'm not considering'.
All that Keltham can really say, in the thirty seconds remaining according to his watch, is that in the end people don't usually assign an explicit probability there. They steer by the relative odds of those models they actually have of the world. And also put some quantity of effort into searching for better hypotheses, or better languages in which to speak them, proportional to how much everything is currently going horrifyingly wrong and how disastrously confused they are and how much nothing they try is working.
And also you'd maybe adjust some of your probability estimates towards greater 'entropy' if anybody here knew what 'entropy' was. Or adjust in the direction of general pessimism and gloom about achieving preferred outcomes, if you were navigating a difficult problem where being fundamentally ignorant was not actually going to make your life any easier.
Keltham: Well, that's it on the Communal Share Language (Baseline). Time for lunch. Not one of his best lectures ever, but not all of his lectures can be.
He did have three extra hours left over on that last Communal Share Language, though, and gave those extra hours to Korva Tallandria, Willa Shilira, and Alexandre Esquerra, if they happen to want to ask him any questions over lunch that could benefit from their being able to still speak Baseline, or if they want to go off and think on their own or review their notes in a way that uses Baseline. They have priority on sitting next to him if they so choose.
There is a discipline out of dath ilan of learning to optimize reality first and appearances second. Yes and indeed, appearances cannot be neglected in human interactions, especially commerce. But if you want to look competent, one should first ask 'How can I actually become competent?' and then 'How can I signal that real competence in a way that's hard to fake?' Any other pathway is one where you're just getting into an arms race between trying to fake how things look, and other people knowing that and not being stupid and guessing what you're faking.
At least, so it is in dath ilan. In Golarion, Keltham gets the impression, there are more complicated things going on. Complicated silly things. It's not going to be like that around him.
Yeah, asking your stupid questions can make you look stupid. It doesn't make you actually any more stupid. Asking your stupid questions does, in fact, make you actually more competent. And benefits Keltham in his teaching, by giving him any idea of what anyone is thinking.
Keltham is not particularly under the impression that the people who didn't ask any questions, if he could read their minds, were not keeping many equally stupid questions to themselves. If he could read their minds, he would update in a predictable direction, so he is just going ahead and updating that way now; he is reminding his emotions as well as his thoughts to update that way.
Whatever it is that would make silence somehow seem like more a good look in regular Cheliax, in a way that's a stable equilibrium of incentives - maybe because what you're really showing off is your prudence or your self-discipline? - it's not a signal that Keltham knows how to read off from their quiet and their controlledly cheerful expressions.
Welcome to Civilization. The equilibrium of your incentives here will not be silence.
Asmodia: ...okay, there may be a problem here where the most vocal question-askers were thus selected to be bad at being a normal Chelish person.
Carissa Sevar: "Proposal for next time: we oblige everyone to attend class with shadowy hoods that conceal their faces and distort their voices, so they can ask stupid questions in perfect anonymity."
Keltham: "Motion to experiment immediately passes, on grounds of value-of-information plus how awesome it would be if we just ended up doing that all the time."
Alexandre Esquerra: Unfortunately, Alexandre (as he heads off to lunch) now needs to focus more on being alter-Alexandre and less on learning useful things.
He would like to know if he can ask Keltham about computer science at lunch, since that is apparently a dark and deadly power that he therefore deeply desires to possess. (He thinks alter-Alexandre still wants to possess dark and deadly powers, even if less because they are dark and deadly than because they are powers.)
Asmodia: Okay, you know what, actually, NO, Alexandre does NOT get to ask about 'computer science'. That is going to be Keepers-only until they know what the hell that is, and by 'Keepers-only' Asmodia actually means 'Asmodia-only', with the transcript going to the Most High only, unless and until either Asmodia or Aspexia Rugatonn says it is fine for other people to know about.
Because alter-Cheliax is a very sensible place about these sorts of things, and if it wasn't before, IT IS NOW OKAY.
Carissa Sevar: That kind of sounds like it might change a lot about alter-Cheliax but it's fine as a specific directive for this project given how much god-bothering it has already attracted.
Alexandre Esquerra: Acknowledged, (- oh no what's her fancy title - most of his best options sound like mocking her - Lady Wall-Keeper would just sound absurd -) Senior Researcher. This slave of Asmodeus will obey your commands.
Asmodia: ...she is starting to be seriously worried that maybe in Golarion weird people are going to be first and best at mastering Law, which has in hindsight been something of a noticeable trend on Project Lawful even before this, possibly because failing at Law is normal in Golarion and anybody who doesn't fail therefore isn't normal, all of which could potentially lead to things getting very odd around Cheliax.
Korva Tallandria: Korva still has enough of a functioning brain to figure out that being explicitly invited to sit with Keltham for lunch is probably a good sign and not a bad one, especially since the offer was also extended to Willa, who she's pretty sure is the best at math in the new group.
This does nothing to change the fact that her emotional response to this invitation is the desire to scream and cry like a literal infant, which even a five-year-old knows much better than to actually do. She wants to go back to her room, bite herself until the desire to cry goes away, figure out how to obtain full transcripts of this lesson, and try working through half the problems again at a pace that doesn't make her feel like she's dying. She also desperately wants to stop having immediate reasons to think in baseline, where all the words are too short and interacting with the normal speed of her thoughts in weird ways, and a bunch of them don't seem to translate into anything except particular technical concepts that she doesn't have enough of a handle on to use.
But she doesn't scream, because she's not an infant, and she doesn't excuse herself back to her room, because she doesn't want to make her weaknesses any more obvious to the rest of her competitors. So she smiles pleasantly, digs her nails into the palms of her hands, goes ahead and eats lunch with Keltham, and hopes that Alexandre and Willa have enough questions to cover for the fact that she's having a horrible day rather than a good one.
Willa Shilira: Willa's so full of questions about everything that happened during the lecture. Usually she would go hide in a corner and eat her lunch alone, but this was like, practically a handwritten invitation! Even alter Willa would have to be super excited about this right? Even shy, hesitant, alter Willa is studious, she couldn't possibly pass up this opportunity.
She's really thankful that there isn't a voice in her head overruling her talking to Keltham at lunch; being alter Willa doesn't seem like it'll be so bad after all! She'll try to avoid any dangerous topics about Cheliax or related things, but that just means she can talk about SCIENCE(!) instead.
Her head is spinning with all the little details and digressions of lecture that were missed or had to be glossed over for time, but there's one question she has to ask Keltham first, she couldn't possibly resist this one, it's just too tempting. Maybe he even intended them to get insatiably curious.
"What's 'computer-science'? Can you tell us all about it?"
Asmodia: Perhaps this would be a good time to dust off Ione's old idea about how one of the most promising new researcher-candidates should get a Fox's Cunning boost and then commit suicide and refuse to be resurrected! They could actually kill Willa too, and not resurrect her, to make that part more realistic!
Alexandre Esquerra: Alexandre refrains from laughing with horrible, malicious glee primarily because then he would be tortured for hours, and secondarily because he is too busy staring in awe, and tertiarily because he has the self-control to get on Project Lawful in the first place and so is maintaining Chelish Dignity. This is great, and also, he's going to be tortured for hours.
Korva Tallandria: Korva, who has the highest Knowledge (Random Nonsense) in the class, has a guess, based on what the words "computer science" and "programming" sounded related to when she first heard them. There are words for the idea of controlling constructs or machines without the use of magic, and Korva has run into them in passing, in a dusty novel that her library had probably forgotten about. It has not yet occurred to her that some people might not have any non-baseline words for this idea.
"Machine control stuff, right?" she says, idly moving her food around instead of bothering to look up at anyone else in this conversation.
Keltham: "Among other things, yes, but I wouldn't have thought you had enough information to reach that conclusion!"
Korva Tallandria: "It's just what the word sounded like, I didn't deduce it or anything."
Keltham: "That is... interesting. I would not have expected 'computer-science' to sound like anything, if it was math you didn't otherwise know. Willa, Alexandre, what if anything did 'computer-science' sound like to you? Try not to let your answer be influenced by Korva's answer, if possible."
Asmodia: (Honesty authorized, if this is going to happen anyways they shouldn't risk alterCheliax over the details.)
Willa Shilira: "It sounded like something mathy to do with discrete numbers changing back and forth?"
Alexandre Esquerra: "It was difficult to comprehend, not easily connecting to anything I understood; something to do with mathematics would be my best guess? There's seems to be some type of connection to 'programming', which translated as - an element of making magic items? Shaping conditionals, or if/then/else statements, into magical items."
(Alexandre has just been ordered to answer, and so does, but right now his top priority is trying to figure out if alterAlexandre could accomplish actual Alexandre's goal of changing the topic of conversation so that he doesn't get tortured later for failing to salvage the situation, or not. He suspects not; he's been issued strict instructions not to do anything alterAlexandre wouldn't do, which means he cannot change the topic to metallurgy or chemistry, topics of much more direct interest to alterAlexandre, because alterAlexandre wouldn't want to offend Keltham by changing his topic of conversation...
He thinks. If his superiors have alternate orders, he is prepared to obey.)
Keltham: "Huh. Share Language might be doing more useful work than I thought, if you're getting connotations like that."
"Fundamentally, 'computers' are - containers of raw causality, raw form, that a 'programmer', shaper-of-raw-form, can put into correspondence with real things or mathematical objects that you want to gain information about. If you got any Validity off the transcripts, it was among the first things I taught here as the underpinning of what Law is at all, if you remember how 'axioms' pinned down the forms of numbers and implied the truths of arithmetic... well, um, either way."
"A 'computer' is something where you could start from YESes and NOs and connect them up by very simple rules like 'YES if the inputs are both YES' or 'YES if exactly one input is YES' and turn that into something which would multiply out 345 * 678 for you, so you didn't have to solve it by hand. Which you'd do by shaping and connecting up systems of YES and NO that evolved over time, by rules that mortals specified, into a shape that mapped your intents about 345 and * and 678 into patterns of YES and NO, and evolved them into a new pattern that represented the answer to that, which I don't have memorized, in the same code and mapping."
"It's not, to be clear, that 'computers' are directly working with the underlying stuff of reality underneath all molecules and atoms. You could make red and blue marks on paper, to represent YES and NO, and operate the rules yourself for how they evolve over time, and exactly duplicate what 'computers' do. 'Computers' just use much tinier marks and do a lot of it very quickly. They can multiply millions of numbers in a second."
"'Computer-science' is how to do clever things by adding and multiplying and branching millions of numbers in a second. And that is, in fact, how Civilization controls most complicated machines, and also how we end up talking to people on the opposite side of the world, without magic."
"The part where you directly work with the underlying 'realityfluid' beneath all molecules and atoms is 'quantum-computing'. Though Civilization was just getting started on that. 'Quantum-computing' would let you do calculations that no ordinary computer could complete before the universe burned out, or get the results of calculations that were only actually performed inside a tiny subuniverse much less real than the universe that gets to look at the results -"
Halfling slave #958245 "Broom": "Keltham. A brief word."
Keltham: Keltham casts Message.
Is the word possibly going to be 'shut up'?
Halfling slave #958245 "Broom": Yes.
Keltham: K, he can talk about this with Broom later.
Asmodia: Apparently there's one other sane person on Project Lawful. Whom Asmodia had almost forgotten about, which, to be fair, Broom is good at getting people to do.
Willa Shilira: Willa thinks that maybe she has never wanted anything as much as she wants a 'computer'. Maybe not even a headband. Building all the way up from working with single YESes and NOs to lots of big numbers sounds really exciting, and then when you were done you could actually multiply the big numbers too, as a bonus! She's about to ask if there's any way they could start making computers, maybe with liberal amounts of magic to help.
And then there is a surprise Broom and Willa starts back ever so slightly in surprise, her mouth snapping shut before she says anything.
Keltham: "Sorry about that. Anyways, 'computer-science' is liable to be pretty far out of reach of what Golarion can do for a while, unless magic is a lot better at manipulating very tiny things than I've heard about so far. If magic can manage manipulations on even a thousand times that scale of tinyness, I can probably parlay that into reversible male contraception, for example."
Korva Tallandria: WHY IS EVERYTHING THIS PERSON SAYS MADE OF MATH.
...that's a very stupid question, isn't it. Korva is beginning to suspect that when you get to Hell, they scoop all of the words out of you and replace them with math, and then every time you talk you have to translate it out of your native language, which is now math, and that the core reason Keltham is valuable is that no one realized that you could do that to someone without magic. But someone did it to Keltham, at some point, and now he's trying to do it to them.
...which means she should try her best to let him do that, but out of concern for her ability to make it through this conversation, Korva is going to try to redirect him away from math anyway.
"That would be cool, but no surprise that it's beyond us for now. As far as share language for lectures goes, I kind of expect there are pros and cons? I'm sure I picked up some things I wouldn't have picked up from a lecture in Taldane, but I think it also got me tripped up in places. I'd try to work through the connotations of some word I only had a vague sense of, and I'd end up unfolding it out into some long Taldane phrase before I felt like I maybe had a handle on it, and by the time I was done I'd have missed a sentence or two and be struggling to jump back in."
Keltham: "Oof. Yeah, it's plausible that literally slowing down and pausing between sentences is a sort of thing I should do in that situation. Hopefully Security can get you a copy of the transcript, I should be reading all of mine and checking them but I do not, in fact, seem to be doing that."
Asmodia: Oh NO oh NO there's no way Korva could have seen that coming but 'Don't nudge Keltham into reviewing all his transcripts' is on the wall in PURPLE and now they need a subject change a SMOOTH subject change that doesn't make Keltham notice anything strange is going on, or some other person coming by to interrupt, or failing that even an outright DISTRACTION quick somebody think of SOMETHING -
Thellim: Meanwhile in a totally different continuum, Thellim is having something of an argument with a man of the planet Earth, whose job title translates as 'scientist'.
(Continued in to earth with science.)