Keltham: Eventually Keltham does succeed in deriving (this time using dubious infinitary arguments, instead of clear and simple combinatorics) that indeed:
If you start out thinking any fraction of LEFT and RIGHT between 0 and 1 is equally plausible on priors, and you see experimental results going LEFT on N occasions and going RIGHT on M occasions, the prediction for the next round is (N+1)/(N+M+2) for LEFT and (M+1)/(N+M+2) for RIGHT.
Keltham: Worth pointing out explicitly:
You could see this whole fragment of mathematics, of Law, as something that learns. If you show it 100 examples of balls with 70 going LEFT and 30 going RIGHT, it'll start to predict future probabilities of LEFT and RIGHT around 70% and 30%.
It only learns a very restricted class of things compared to entire people. It can never learn, if you show it the sequence LEFT LEFT RIGHT LEFT LEFT RIGHT repeated 20 times, that it should predict LEFT with very high probability on the next two rounds, followed by RIGHT with very high probability. It'll just predict LEFT with probability approaching 2/3 and RIGHT with probability approaching 1/3.
This, of course, is because people contain much more complicated and powerful fragments of Law within them, enabling people to learn much more complicated and powerful fragments of reality.
Even dath ilan doesn't know that much Law.
It's the knowledge a god would need to build a mortal from scratch.
dath ilan: (Certain parts of Civilization do, in fact, secretly know that much Law. It's unfortunately only one piece of a much larger, insanely lethal challenge.)
Carissa Sevar: - well now she wants to know whether the gods know how to build mortals from scratch.
Not the point, probably.
Keltham: Keltham is having similar thoughts! He'll wonder out loud about whether Golarion's gods know enough Law to do any significant editing? If the newcomers haven't read the transcripts for this part yet - or it was less clear from the transcript than in-person - Golarion humans were obviously copied off some ancestors or cousins of dath ilan visible from Golarion's multiverse, the same way Keltham himself was able to appear here. Keltham can eat Golarion food and that doesn't happen unless the proteins inside are the same and that requires common ancestry. Similarly, the humans here birth half women and half men, so nobody tampered with that part.
If anything Keltham's sort of surprised that people here aren't more edited. He wishes he knew what the gods' original edits to mortals were, whose reversion or failure was described as mortals gaining "free will". That would provide a lot of hint about what level of innate-Law-editing the gods were capable of deploying.
He's guessing it was relatively shallow, though? Not really anywhere near the level of 'make a mortal from scratch'? Humans here wouldn't have shared ancestry with dath ilan if gods here could just make mortals from scratch. The fact that the gods' edits eventually failed, and that people basically turned back into humans afterwards, says that it was probably much more like imposing extra structure on top, maybe magical structure. Not really remaking and rebuilding things like a 'programmer' could do inside a 'computer', probably not even 'compiling' new 'genes'.
Well, that seems to square with the rest of reality as Keltham observes it? None of the gods here seem powerful enough to be freely wielding Law on that level.
Korva Tallandria: Korva is pretty sure that the gods did create mortals from scratch at some point - even with the claims about heredity in some of the past transcripts, which she has read, she's still extremely dubious about the idea that no higher power has ever shaped a lower one from nothingness or near-nothingness - but it does seem possible that no mortal has ever developed the power, and if it can be done with math at all, then it makes sense that it's the sort of thing a mortal could do. If Keltham does know some sort of building block towards mortals shaping mortals from scratch, that's the sort of thing that would go some way in explaining why the gods are so obsessed with him.
It's terribly inconvenient that this would actually be an important thing to study, given that Korva hasn't been following along for the past several minutes and is once again despairing of her future. Although if even dath ilan hasn't figured it out, then it seems pretty improbable that the collection of people here will.
...maybe she can at least study the history of beings whose creation might have been directly observed or heard of secondhand, and whatever is known of the changes that occur to humans after death. There. Not useless.
Carissa Sevar: Thaaaat's going to have people quiet, nervous of heresy. "There are other races that are different from humans but similarly patterned, so on your theory I imagine the gods copied the shared-ancestry that also produced dath ilan and then made changes to produce, say, elves, or halflings."
Aaaaand probably don't list the very large number of other species that are people, lest that get Keltham thinking about fire elementals.
lintamande: "It's said that the First World is what the first attempt at creation was like, so you might study the fae if you were trying to understand how the gods shaped humans," Meritxell says. "Aside from the thing where you definitely should not study the fae."
Korva Tallandria: - you know, actually, also, who's to say that dath ilani weren't copied off of Golarion humans and then sequestered in some place as some random guy's experiment, like OKAY YES SHE'S THINKING ABOUT HERMEA, given that Golarion obviously has more planar and possibly even more interstellar reach and there are various people who would absolutely do this, and that's why dath ilani apparently don't know anything about their history. Egotistical, much?
Keltham: "Yeah, there's an awful lot of things in Golarion I'd run off and go look at, if it was safe for me to leave my fortress, and I didn't have more critical-pathy things to do with my time."
He does grok the concept of a critical path, though, that is also common wisdom out of Civilization for people building startups. At some point he may start to develop cabin fever, but Day 20 is not that point.
Ione Sala: "Your regular reminder that I will do anything, to go along with you at that time."
Keltham: "Yes thank you I'm aware. But as with sexual attraction of you towards me, my own desire to have you follow me around forever, telling me not to do things, may also take some time for me to develop."
Ione Sala: "Okay, just saying, you will die in ten minutes if I'm not there and you will somehow manage to get your soul stuck in the process."
Keltham: "Ione, I'm not saying no, I'm saying our 'moirallegience'* has not yet progressed to that point."
"Anyways."(*) Not the literal Baseline term.
Keltham: By way of finishing up the morning's lecture, Keltham will also explicitly spell out the point that - outside of particular metahypotheses you're deliberately reasoning inside - it's generally considered a bad idea to assign Probability Absolutely Zero to anything.
Like the way that the Rule of Succession metahypothesis implicitly assigns Probability 0 that the ball just always goes LEFT or always goes RIGHT, because then you can never ever learn that thing, no matter how much evidence is presented you. The Rule of Succession never figures that out for real, no matter how many cases of LEFT it sees.
If you assign Probability 0 that...
Good examples of somebody-might-think-they-were-probability-0 statements are harder for an outsider to come up with when they're relatively new to Golarion.
If you assign Probability 1 that grass is green, and then you go outside and suddenly all the grass looks blue to you, well, obviously you just conclude that your eyes are deceiving you. Sure, you're vastly more likely to see the grass looking blue, if in fact the grass looks blue. Maybe it's a million times more likely. But it's not infinity times more likely.
If you say there's Probability 1 that grass is green, you're saying that's infinity times more likely than any hypothesis where it's not green. And the evidence of your senses is always finite, they could always be mistaken. So if you have an infinite prior, that's like being something that can't notice the truth, ever, no matter how hard and how many times you're hit over the head with it.
Thankfully, human beings are just more complicated things than the sorts of hypotheses that assign Probability 0 to the ball always going LEFT. Even if your conscious deliberation somehow gets wedged into weird states where you claim to be infinitely sure of things - which, like, Keltham finds it hard to visualize anybody thinking that could possibly be a reasonable thing to do in the first place, like, how would you ever be able to Lawfully justify a claim to be able to guess something strongly enough that you could make guesses like that infinity times and be wrong zero times - but they did teach Keltham not to do that, so it's probably a sort of thing that people ever do, especially when they're kids -
- even then, a human being is a more complicated thing, and there'll be metahypotheses inside you that some part of you is still processing, still able to learn the thing that you claim to assign Probability 0.
If you say to yourself that you believe absolutely that the ball can only ever have some propensity to go LEFT or RIGHT, independently on each round, and any fraction between 0 and 1 is equally probable - and then reality shows you LEFT RIGHT LEFT RIGHT repeating forever - there will still be some part of you that notices.
Because you're not actually a kind of thing that can really assign Probability 0 or Probability 1 to anything, in all of your parts, just by saying that's what you're doing.
There'll still always be some part of you that can learn from sufficiently blatant evidence, that you're hit on the head with sufficiently repeatedly. If it's a sort of thing that humans assign probability greater than 0 - equivalently, if it's a sort of thing that humans can learn - in the parts of you that your conscious decisions can't cripple enough.
This was also told to Keltham when he was a child, and emphasized to him. He's not quite sure why it's important, no sane person would assign Probability 0 to any interesting proposition in the first place, but it sounded like a warning so he's repeating it.
Carissa Sevar: That feels - disquieting, somehow. She's not sure it is something to be thankful for, and she's not at all sure it'd be true of devils.
Pilar : "1 + 1 = 3?"
Keltham: "Drugs, talk-control, hidden superbeings who created you from scratch to be mistaken about that. In Golarion, Suggestions and more classified forms of mind-control magic, gods directly messing with you, hidden superbeings who could mess with gods."
"If one day you started adding 1 and 1 and getting 3 every time, including for things like physically adding an apple to an apple in a basket and getting three apples, you would eventually decide your old memories had been mistaken. It's a kind of pattern you would be able to notice, if it were true. So the probability that all of yourself assigns to it isn't 0."
Korva Tallandria: You know, some of these warnings do actually make more sense as things to specifically emphasize to people if dath ilani education actually was created by some process that was specifically reacting to things that had been seen on Golarion. Just saying. Well, thinking.
Pilar : Some of these emphases would make more sense if the dath ilani Keepers were specifically trying to counter Asmodeanism. As taught in Golarion.
Maybe - faith in gods in general? Pilar is less sure about that part.
Keltham: "And on that note of irreducible epistemological existential terror, let's break for lunch!"
Asmodia: Well, that was one of the most unnerving mathematical experiences of Asmodia's life to date.
Otolmens: Otolmens was also unnerved by some of that so-called MATH.
The fact that it WORKED is even WORSE!
Mortals using INVALID reasoning to arrive at CORRECT answers may be all too close to using VALID reasoning to produce WRONG answers.
Project Lawful: PL-timestamp: Day 24 (20) / Afternoon
lintamande: When it became obvious that Keltham would benefit from an alchemist Cheliax started reviewing candidates. Alchemists, of course, don't tend to advertise all their breakthroughs; that's a great way to inspire people to send spies and steal all your hard work and secret recipes. If you drop by every alchemists's workshop and ask them how much they have independently catalogued properties of gases and metals, the floor will fold under you and drop you into a pit of acid.
If you have been urgently dispatched for this task by the Crown, you presumably have a Fly spell active, and you can remain in place and present your credentials and ask again, less politely.
By the day after Keltham asks for an alchemist they've screened some, and picked one, and packed him off to Ostenso with a Bag of Holding stuffed with several thousand of his acid-damaged notebooks.
He is carefully not looking at anything in particular about this place, in the hope that this'll make it less necessary to kill him afterwards.
Keltham: This is Keltham! He seems like an obvious mad-experimenter alchemist apprentice though slightly off in several ways. He comes from a world that has zero knowledge of magic and far vaster knowledge than Golarion of everything that isn't magic, and he's going to use that knowledge to cheaply produce vast quantities of distilled oil of vitriol and drop spellsilver refining costs by a factor of ten! They're rapidly advancing in applying his otherworldly knowledge to Golarion!
Today's frontier of research: Recreating an otherworldly instrument that will burn materials in flame and send the resulting light through a crystal, to show the exact colors of light inside, to determine what fundamental constituents are within nonmagical matter! Is that a sort of thing that'd be useful to alchemists in general, by the way?
(Elapsed time until he acquired knowledge that Cheliax would kill him over: 47 seconds.)
lintamande: Well. That's not great but he'll just have to be very useful to this project so he's kept around for it.
That....is indeed the kind of thing that'd be useful to alchemists. In general.
Keltham: Keltham is continuing to talk rapidly, obviously excited about this meeting! He is pleased to be introduced to what is, by his standards, an otherworldly mad scientist much like himself! Has the alchemist done a lot of experimentation on human subjects, by the way, and if so, what kind?
lintamande: - yes, mostly for potionmaking rather than metal refining or distilling? He can't actually think of a useful human experiment he'd run on refining or distilling but for potionmaking you need human subjects for testing dose, testing effect duration, testing toxicity, testing side effects, and so on, it doesn't extrapolate reliably from rabbits.
Keltham: Interesting! What's the most toxic thing he's found so far that way?
lintamande: - he's not actually a poisons expert and doesn't work with anything as lethal as, say, bluering, his potions experiments have mostly been on concoction miscibility - the known problem where taking multiple alchemical concoctions simultaneously has hard-to-predict side effects, some of them deadly and some of them actively desirable, so if you could only figure out how to reliably get the desirable ones you'd be very wealthy selling to adventurers. Unfortunately minutia of dosing that vary person-to-person and day-to-day seem relevant to concoction miscibility, and of course test subjects tend to expire of mistakes, so it's a difficult area to make progress in, but worth it when you have the subjects on hand.
Keltham: The government doesn't require him to buy resurrection insurance or anything on them, does it?
lintamande: That is the most baffling question he has ever been asked and he does not like this entire line of questioning, in retrospect. "- depends. Is that how it's done on your originating planet?"
Keltham: His home planet doesn't have resurrections. No magic, remember? They put people into very deep cold instead and will do something with them later.
You can get people who'll let you kill them in alchemy experiments, but they're very expensive, obviously.
What does the need for resurrection insurance depend on?
lintamande: - uh, sometimes your experimental subjects are old or sick and going to die anyway. And in that case you don't need resurrection insurance.
Keltham: Uh huh. And how much do subjects like that cost?
lintamande: ....fifty gold?
Keltham: That does sound cheaper than resurrection insurance! Much cheaper! Makes you wonder why he'd use any resurrection-insured subjects at all, really.
lintamande: .....well, it's much harder to get those subjects.
Keltham: Uh huh. So they're much cheaper than resurrection insurance, but much harder to get.
And how much does resurrection insurance cost for him?
lintamande: ...it depends a lot on what specific experiments you plan to run.
Keltham: Whatever he was working on last week will be fine.
lintamande: - he wasn't working on this last week! He doesn't do that much with admixtures! He thinks a couple years ago when he last had a theory worth testing it was ...1000gp?
Keltham: Keltham must have really not liked that answer! His whole face starts to change.
Asmodia: "You failed at the point where you told Keltham that buying killable test subjects is much cheaper than resurrection insurance on them, but also they're much harder to find. If that had been true, their price would have gone up until the supply increased and the demand went down."
Asmodia has now read an Abadaran book!
lintamande: "All right, in this game of yours how expensive are test subjects? And do you have to pay to resurrect them?"
Asmodia: The girl who looks like and dresses like a second-circle Ostenso wizard academy student is speaking very levelly.
"This is not a fucking game, alchemist. You just failed a live-fire test, one you thought was real. The only reason you're not in a lot of pain right now, is that we do not expect putting that much fear into you to help you keep your composure around Keltham. Project Lawful has higher priorities than the Asmodean condition of your soul."
"But if you'd actually fucked up, tipped Keltham off, if he'd ended up in Osirion constructing weapons for them to use against Cheliax and that was your fault - you would have personally pissed off, among others, the Queen of Cheliax, a Count of Hell who bought my soul on the speculation that Keltham would make it more valuable, and, probably, Asmodeus Himself."
"Keltham has not, so far, probed any of the outside experts brought in. He has asked questions like that of Project employees, which is what we are to him. If you are sticking around then he will probably ask questions like that of you at some point. You're either ready for it when it happens, or you are a liability who should never be introduced to Keltham in the first place."
"50 gold pieces for a death volunteer didn't sound impossible to me, given that slaves with a lot of lifespan remaining cost 75gp, but that slavery is mostly illegal in alter-Cheliax. But if that's true then you only use dying subjects and never buy resurrection insurance on healthy subjects. Everything about alter-Cheliax MUST be consistent."
lintamande: "How about I just don't deal in human subjects at all."
Asmodia: "Are there alchemists like that in Taldor?"
lintamande: "I would assume so, what with how human subjects are expensive and irrelevant for most of what we do!"
Asmodia: "Then we may be able to get away with telling Keltham that lie."
"Why don't you work with human subjects? You don't come across as particularly squeamish in conversation and attitude. I suppose I've never actually met somebody who is, but Keltham will have."
lintamande: " - because it's a dead end? People have been trying to crack potion miscibility for as long as there's been alchemy. The first person to get it will be richer than the gods. But if you buy a hundred fucking slaves and feed them all your perfectly optimized concoction adjusted for their body weight and sex and astrological sign and hair texture and whatever else you thought of, you'll get a hundred different results, and five of them will drop dead, and if you try repeatedly then everyone'll be dead by the end of the month. It's worth a try whenever you think you've thought of something new that it's possible no one else has found, but it's a stupid waste of time and money to try to build a career off. The interesting progress in alchemy is all in ores and oils, and rabbits are cheaper than slaves for practically all testing."
Asmodia: "Is that what a Taldorian alchemist who doesn't work with human subjects says? Because that sounded at least a little Chelish to me. Especially the part about slaves."
lintamande: "If you pay a hundred elderly volunteers, they'll all be dead by the end of the month. If you want an alchemist from Taldor why didn't you hire one of those, I've never been to Taldor."
Carissa Sevar: "It seems like they'd make that difficult for us. Try it again, with an Eagle's Splendour, see if that helps your bluff any."
Asmodia: "I think we also bring in the fake paladin. I have a sense that some of this would have twigged Keltham's sense for - real Evil, not what we've told him is Evil - and that's something nobody else here knows how to fake caring about."
Carissa Sevar: Carissa gestures at Security to do what Asmodia said. "How do people become an alchemist, how did you become one?"
lintamande: "I was apprenticed to one, and I was the least incompetent of his apprentices - do they have apprenticeships in Taldor?"
Carissa Sevar: "Yes."
Asmodia: Asmodia has a nervous feeling - as expressed within her state of continuous low-grade panic - that at some point Project Lawful is going to fuck up on account of nobody on staff being really actually Good, knowing what that's like inside instead of outside.
Not something that could be fixed easily, or at all, in any obvious way. But just because you can't fix something doesn't make it be not a problem.
...why is her brain suggesting that they ask Pilar? Pilar's curse isn't that frequently helpful and couldn't be trusted anyways.
Project Lawful: PL-timestamp: Day 24 (20) / Mid-Afternoon
Keltham: Keltham returns from having taken a couple of hours' break for wizard practice, scroll practice, and walking around outside but still within the Forbiddance.
On his return, he finds... they've got an alchemist for him already! Apparently somebody saw this one coming and started the Security screening process earlier. Good for them! Well, time to introduce himself and his work, then!
Keltham: This is Keltham! He seems like a pretty extreme mad-experimenter alchemist apprentice, the sort who's obviously going to end up dead within weeks if not days. He's not just slightly 'off', everything about him is way off in more directions than you can easily count.
Today's frontier of research is a spectroscope, which burns matter and uses a prism to refract the light into component colors, or shines light through the burning material in order to check for absorption lines within the colors.
Why is that important? Well, light is made of particles called photons, the color corresponds to the energy of the photons, and the emission-absorption lines correspond to increases or decreases in energy of the 'electrons' that are constituents of the 'atoms' that are the almost-perfectly-stable constituents of matter. The electron orbitals are a huge determinant of how matter behaves chemically, how it gets transformed by acid or what other atoms it combines with into semi-stable molecules.
Part of their major project here is to see whether informed use of Prestidigitation can greatly change the reach of what's possible in chemistry. Prestidigitation can change color, stickiness, taste; these are all things that plausibly depend on electron orbitals. They've already verified that Prestidigitation can change the reaction rate between acidic vinegar and basic wood-ash lye. Lady Avaricia was first other than Keltham to Prestidigitate matter such that it would then burn with a different color of flame; Korva Tallandria managed to get matter to not only burn with the color of sulfur but to smell something like sulfur and to produce coughing like sulfur dioxide.
Keltham is hoping that by using a spectroscope on Prestidigitated matter, his researchers will be able to get direct feedback on how their Prestidigitation is changing the behavior of electron orbitals. That seems like it could be one of the key steps in learning to make elements behave like other elements that Keltham doesn't have time to identify in known Golarion materials, or even behave like elements that can't exist as ordinary matter. It will enable them to regularize the transformations that Prestidigitation applies to materials, even if two different researchers think copper tastes slightly different to them.
Keltham is mentioning all this, of course, because he's sure that the alchemist is wondering why, if you can use Prestidigitation to control chemistry that easily, nobody's ever done it before. They've had some indication that you may need to know something about electron orbitals, before people seem to be able to conceive in the right way about which transformation they want to apply, by controlling the taste of something, that would also make it burn a different color - his researchers weren't able to do that right away; as of yesterday, some researchers still couldn't. But also, Keltham is guessing that it would be a great deal more difficult to map your way through Prestidigitating chemistry if you didn't understand the underlying atomic interactions, started out by trying to Prestidigitate complicated tastes into other complicated molecules instead of atomic tastes into atoms, and couldn't use a spectroscope to see what your early attempts at Prestidigitation chemistry were changing about the electron orbitals.
Besides making distilled oil of vitriol and refining spellsilver, are there any other non-magical chemical transformations that would be hugely profitable to improve or that would be vital to the economy of Cheliax or other countries?
(Elapsed time before he learned knowledge that any government in Golarion would kill and soul-trap him over: Three and a half minutes.)
lintamande: - yes, yes, there are some. For nonmagical reagents, the ones used at large scales where it'd be important to invent a cheaper version are bleaching powder, used in cloth production, and soda ash, used for glass, textile, soap, and papermaking. There's a standing large prize for a process to produce alkali from sea salt; likewise there are efforts to make naphtha, pitch oil, and sal ammoniac.
Keltham: Keltham doesn't have memorized that many processes for producing chemicals at scale, alas. He knows one bulk process for producing an alkaline substance, the same way he knows a process for bulk-producing the most industrially important kind of acid. Lady Avaricia is trying to identify Golarion materials in terms of the underlying constituents of matter that Keltham is familiar with; she'd be the one charged with figuring out what 'soda ash' and 'bleaching powder' might be in his terms.
Oh, also the alchemist shouldn't take it personally if Lady Avaricia thinks he falls far below her standards or seems rude. Lady Avaricia is one of those people who can't bear to conceal outward appearances, and thinks that everyone and everything is terrible; it's nothing personal if she thinks it about him, and says so out loud.
What sort of incredibly finicky processes are there for producing small amounts of very expensive things, such that substantially higher demand would exist if the cost decreased? Spellsilver itself is one of those things from Keltham's viewpoint; it's being produced in very small quantities via a finicky process that Keltham hopes Prestidigitation can smoothen and regularize, and there'd be demand for much higher quantities of spellsilver if those were available.
lintamande: This is so much better than the conversations with fake-Keltham. He can name some candidates.
Keltham: If there's one of those that has relatively less expensive inputs and is just labor-costly because it's hugely finicky, the Project might want to practice on that. And if there's none of those with expensive outputs, they could try perfecting via Prestidigitation some processes that sell for less. If his researchers maybe want to mess around with it whenever. Rather than, you know, the boiling-acid version that should only be done in groups, and with Resist Energy cast. Keltham has given them some known chemical reactions to mess with, to see if they can steer around reaction pathways, but those reactions are pretty elementary and not that finicky to start with; there'll be some different game for mastering the trickier processes.
Their current problems along the way to completing a distilled-oil-of-vitriol production step, include:
- Figuring out if they need to purify oxygen for step one, where they burn sulfur to turn it into sulfur dioxide, or if regular air works. If regular air doesn't work, they'll need to liquefy air to distill it; or use a controlled-lightning process on water to make combined Element-1 and Element-8 gas.
Also, it would be helpful to know if any of the "create a bubble of breathable air" spells work in a way where they can extract purer regular air from that, which will be useful in chemical reactions.
- Either figuring out some known Golarion metal is element-23; or getting something else to act as 23 for purposes of its oxide catalyzing SO2->SO3; or directly Prestigitating SO2 in a way that makes it easily combine with oxygen at high temperatures to form SO3. They also need to know if pure oxygen or at least pure air is required for this step; Keltham suspects you can't just use atmospheric air here, but he's not actually sure.
- Measuring temperatures for purposes of getting the SO2->SO3 furnace to above the melting point of water by 450% of the difference between the melting and boiling point of water. Mercury thermometers don't go that high and are kind of dangerous anyways, unless they're very certain that healing or Restoration works to solve a long-lasting poisoning problem with mercury.
lintamande: Restoration should fix damage from mercury, he can't think why it wouldn't.
(He avoids suggesting a quick human-subject test to be sure.)
He does happen to know that bubble-of-air spells make air, rather than pure oxygen; pure oxygen has been isolated and he has a secret technique for doing so, as well as one for measuring temperature, which he'd be willing to share with this project for half its proceeds.
Keltham: Half the proceeds from what, exactly? Selling the purified oxygen if the Project can figure out how to make his process scalable, and they can't find a better process that's more scalable and not derived from his?
lintamande: Half of the proceeds from selling, or selling the secrets to making, the alchemical products and ores-refined-alchemically and oils-produced alchemically this project is hoping to produce.
Keltham: Doesn't particularly match up with Keltham's model of how hard it would be for him to produce oxygen, or find another alchemist who knew how. There's a lot of stuff with oxygen in it. Water is 8/9 oxygen by weight and Keltham knows how to split it. There's no way that saving a couple of weeks on hammering out the issues in a reconstructed method of dath ilan would be worth half the proceeds of the Project on all chemical sales forever; oxygen simply isn't that important an input.
Keltham has in fact heard about Golarion's concept of bargaining, and could potentially try doing things that way by countering with a ludicrously unfair offer of his own? But first Keltham wants to make sure that he's not missing some key facts that would make this a sincere and fair offer, one that Keltham shouldn't be trying to bargain down from.
Keltham's got truthspells, and also a spell that causes people to only say prices that would be fair given their knowledge, if the alchemist wants a quick way to prove that he's being honest about something or just prove that the entire offer is one that's fair relative to his own model of things.
lintamande: The alchemist doesn't know what Keltham means by a 'sincere and fair' offer. It's sincere in the sense that if Keltham takes the offer he will share all his secret techniques; it's fair in the sense that it's 50-50, which is ....in some sense the fairest possible division of anything?
(Is that what Good people think????)
He knows secret techniques for other chemicals and is willing to teach them too for half the proceeds. Notably alchemists don't usually teach their techniques at all, certainly not to someone who wants to scale them; why, then you'll never make money off it again! You sell the results and keep the process secret. He is only offering to sell the techniques at any price because this project does seem promising.
Keltham: Keltham is not particularly expecting the alchemist's personally derived secret knowledge, relative to more public and commonly known alchemical knowledge, to contribute even 0.1% of what Keltham remembers from an entire alternate world with a vastly more advanced comprehension of nonmagical materials science. As produced by literally millions of alchemist-equivalents with 20+ Intelligence, freely sharing results with each other, and paying 'patentgratuities' on useful discoveries in proportion to how useful they were.
If the 'patentgratuity' methodology was applied to the Project, they'd end up paying him for the two weeks of time that ended up being saved by their using his secret oxygen production technique, or more if it later turns out that it'd have taken longer than two weeks to get lightning-based water splitting going.
The basis on which Keltham had been planning to do this was more prenegotiated though: if the guy proved as valuable as a regular Project employee, Keltham was going to offer him the 0.1% share of the Project that non-elite employees get; if as valuable as an elite Project employee, the 0.2% that elites get. That'll be harder for him to sustain over time, as the younger Project employees learn more new methods and math, but earlier contributions are also more valuable. Those shares of the Project do need to last through all the employees that will prove worthy of them; later employees will get offered more like 0.01%.
If the alchemist isn't happy with either of those, the Project will pay his regular consult fees, not give him a share of the Project, and use only what's public knowledge in Golarion. Keltham assumes the guy has already signed the standard non-disclosure agreement saying that he cannot, without the Project's permission, use or talk about stuff like Prestidigitation chemistry? Also, can the guy use Prestidigitation because if not he may need to spend a week picking that up.
lintamande: 0.1%???? A thousandth? For an alchemy project without an alchemist on staff?
Keltham: Has somebody possibly not explained the scope of the Project to him?
The plan is to use the knowledge out of Keltham's world to revolutionize all nonmagical farming, medicine, clothmaking, mining, metallurgy, manufacture, roadmaking, etcetera, all of which actually scale up much further than Golarion has taken them so far. Probably also use the methodologies out of Keltham's world to revolutionize a bunch of magical stuff too.
Roughly, the plan is to turn Golarion into its own version of his Civilization, as the people of Golarion may want to become that, and the Project is step one of it.
If all dreams fail and the Project ends up being worthless - as does not particularly seem likely at this point to be the case - then the alchemist still gets to keep his usual and customary fees, and the people on the project can sign non-disclosure about whatever secrets of his he revealed to them along the way. Alternatively, if the Project plays out to the extent of being able to capture one gold piece of value created for everyone in Golarion, it ends up worth something like a billion gold pieces, on that theory, and 0.1% is worth a million gold pieces.
This is what the god-war a couple of weeks ago was about, FYI; it started with Zon-Kuthon trying for a decapitation strike on the Project two days after Keltham arrived in Golarion.
lintamande: Right, okay, then.
He does want the nondisclosure agreements in the case the project fails arranged first.
Keltham: Naturally!
lintamande: Also he would absolutely LEAVE if that was POSSIBLE but he's not going to say that.
Keltham: Keltham will work up a nondisclosure agreement shortly, they can stick to public knowledge about alchemy until then. Most of them this afternoon are going to be working on completing the spectroscope as equipment. Can Keltham turn him over to Lady Avaricia, so Lady Avaricia can introduce to him the basic concepts out of Civilization, and he can see if his public knowledge works for helping her to complete her project on relating Golarion concepts to dath ilani ones? Again, it'll seem like Lady Avaricia hates him, but it won't be anything personal, and he doesn't need to be guarded or tactful around her either.
lintamande: - yes, he can do that. Is it her husband's title or her parents'?
Keltham: Parents, Keltham thinks, something about a county she's heiress of.
lintamande: - right. He appreciates the clarification, such as it is, and would be happy to work with the Lady Avaricia on the spectroscope, if it pleases her.
Keltham: Literally nothing's going to please Lady Avaricia until the Project scales to the point of distributing artifact headbands with +6 intelligence, to people who were otherwise nearly the smartest innate prodigies in the world before then, whereupon they may begin to barely meet Lady Avaricia's standards.
Lady Eulàlia Avaricia de Seguer: "Just get over here and assist me already, I don't prefer you standing around."
lintamande: ....great. He'll do that. He's so looking forward to being part of this fascinating project.
Keltham: Message to Lady Avaricia: Guy seemed slightly weird, please observe and report back if he seems even more horribly incompetent than you'd expect from the next-best alchemist to pass their Security screen.
Lady Eulàlia Avaricia de Seguer: She's desperately curious what specifically about that Keltham thought was weird, but doesn't ask.
Keltham: They'll almost get a spectroscope working before Keltham calls it a day!
Project Lawful: PL-timestamp: Day 24 (20) / Evening
Keltham: Martial arts practice. It's getting a little psychologically easier each time, as his mind accepts that injuries to himself and others just are not permanent.
He can't feel about pain what Carissa feels, he doesn't think. But there's a triumph in making it through, walking away healed and less afraid, step by step becoming somebody who's Been In A Fight.
...the thought of fighting Meritxell at all, let alone Meritxell transformed into Carissa, still seems flinchy. He might possibly need to equip Meritxell with a dagger, or something else that makes her look more dangerous, and be attacked by her first, before he'll feel okay about fighting back.
It'll probably go away, that hesitation, after the first time. Things like that are usually like that. Right?
Using Bull's Strength and Cat's Grace to train martial arts is like soaring. Keltham would say, like flying, but eventually he'll be able to Fly - or cast it from scroll - and probably that'll be a totally different sort of soaring experience.
At 4th-circle, Keltham can't sustain those enhancements through a whole training session - nor would he, he should also train fighting unenhanced - but Keltham begins and ends with them, each time. Because elementary hedonics.
Keltham: He'd thought to date Ione, tonight, but ended up feeling not in shape for it, after martial arts practice. It's by far the most mentally strenuous thing he's done repeatedly since he came to Golarion.
He summons Yaisa instead. Tells her to be slow and gentle, to make him feel cared-for, and lets her do all the work.
He should probably talk to Meritxell soon, if she's amenable, fight her soon, and get that part over with.
Project Lawful: PL-timestamp: Day 25 (21) / Morning Lectures
Keltham: All right, Keltham knows this is probably going to be a bit stressful, but...
Keltham: The new researchers are coming up on the end of their trial week tomorrow.
Keltham: Keltham, in retrospect, has not been stress-testing them to the same extent as he stress-tested the current researchers before they got hired.
Keltham: They've now been introduced to the simplest of inductive Law-fragments, the Rule of Succession.
Armed with that, they should be able to figure out how you'd go about deciding whether two different sources of LEFTs and RIGHTs had the same frequency, or a different frequency, if the datasets weren't so different as to make it obvious at a glance.
...and say what it is that an experimentalist out of Civilization would say, and not say, in their report summarizing their data. It's not going to be, "I decided these two sources were the same" or "I've started betting these two sources are different."
...and maybe talk about a certain general symptom of error that might appear, abstracted beyond just the Rule of Succession, if two different experimenters had ended up performing different experiments with different properties, without anyone realizing that earlier, when somebody went through the reports out of Civilization side-by-side.
If they can keep going, and find even more to say, they're welcome to do that.
Split up and try it on your own.
Everyone should have been told earlier this morning to prepare one Fox's Cunning and one Owl's Wisdom; you may use those at will. Researcher-candidates will also get an additional Fox's Cunning supplied by staff.
You've got until lunchtime.
Keltham: Note, this test does not completely determine your final relationship to Project Lawful.
Keltham has been watching them this whole week, not just now. Keltham doubts his saying this will affect Lady Avaricia in any way, so he will mention, for example, that Lady Avaricia could completely blow this one, and still get in on the strength of her chemistry work. People can also end up rising or falling in tier over time, if they're hired.
So don't stress out too much over this one test...
Keltham: ...is what Keltham would like to say, but yeah, realistically, he knows they're going to stress out about it. He apologizes for that part. Last time the researchers ended up not knowing exactly what they were being tested on, and that... was not really a better way of doing things, he doesn't think.
There'll be a predictable two-hour recovery period after lunch, before they start the afternoon's research.
The whiteboard now reads:
Wielding the Law of Succession, that after N lefts and M rights, the odds of LEFT vs. RIGHT are (N+1) to (M+1):
- Figure out how you would analyze whether Source 1 and Source 2 of LEFTs and RIGHTs had the same propensity p for left-vs-right, or different propensities p1 and p2. Assume it's not obvious at a glance.
- Say what an experimentalist out of Civilization would say or not say in their summary of their data. (It won't be, "I decided these two sources were the same" or "I'm betting these two sources are different.")
- Talk, if you have the time, about a general symptom that might appear - not just in analyses using the Rule of Succession - when two different experimenters end up performing an importantly different experiment-on-reality and end up with two datasets generated by importantly different sources, as might turn up when someone out of Civilization was analyzing their datasets or experimental summaries side-by-side.
- You are welcome to say more, if you have more to say.
Project Lawful: PL-timestamp: Day 25 (21) / Morning Test of Doom
lintamande: A stupid thing you could try - it might not work, but it seems like a place to start - is take the first set of outcomes as your starting guess for the second set of outcomes, and then do the math they learned already to adjust the first set for each additional trial, and then you'd end up with your new estimate. ...would that produce the same result if you did it in either direction? It should. So the difference in estimates from each individual estimate to the combined estimate should be equal, if they were tests that ran the same length, which wasn't specified......
...is this really better than eternal torture....
Willa Shilira:Willa's Story
Someone reading Willa's thoughts might be surprised to find out that she isn't afraid, she's excited. Willa's good at tests, the tricky 2-4-6 thing from earlier this week notwithstanding.
And this one isn't like that, it's pretty well defined. If anything's going to get in her way of being special, it won't be a test like this. She keeps her boosts back for now, she can at least see how to start already...
If the first data set has N Lefts and M Rights, then that would inform a set of relative Posteriors:
p^N*(1-p)^M
For some probability p that each ball goes Left.
Then you would want to normalize those to sum to 1 if possible, you could use the result Keltham already got for that because it's a pretty hard problem on its own; she looks in her notes and finds that it's M!*N!/(M+N+1)!. So for the first data set the Hypotheses of left probability have their own probabilities:
P1(p) = [p^N*(1-p)^M]/[M!*N!/(M+N+1)!]
Similarly, if we say the second data set has L lefts and R rights, we can deduce the same probability function there:
P2(p) = [p^L*(1-p)^R]/[R!*L!/(R+L+1)!]
Willa feels sure somehow that the right solution involves using these two functions together somehow. She could instead just use the first one and then feed the other data into it, but why should one data set be treated differently than the other? The situation is symmetric, so the Law should treat them symmetrically.
So what's the important thing here? It's tempting to say that the functions want to be the same, but that's wrong. Both functions could have no data at all and both would be the same flat line, and she'd know nothing at all about if they were the same.
To KNOW FOR SURE the ps are the same, or different, you would have to know p exactly. The function would have to be a lone spike of probability somewhere in each case. Like if P1 was a spike at p=0.5, and P2 was a spike at p=0.6, then you have a 0% chance they're the same. Similarly, if they're both spikes at p=0.5, there's a 100% chance (as long as the model is right in the first place...)
But how do you actually process the P functions to get those 100% or 0% or anything in between chances? Well, what do you always do with probabilties? You multiply them. So then... you'd have to multiply these functions together. With an integral?
Willa's been feverishly learning calculus since she saw it used to such powerful effect, she thinks you'd integrate the two of them multiplied together, and it would be a definite integral. You'd be integrating over the little probability, the p, so dp. The bounds would have to be from p=0 to p=1, the set of possible outcomes.
Would you have to normalize? Scary, she isn't sure. She'll think more about that part later.
INT([p^N*(1-p)^M]/[M!*N!/(M+N+1)!]*[p^L*(1-p)^R]/[R!*L!/(R+L+1)!],p,0,1)
What a mess. But a lot of this doesn't even have p in it, it can seamlessly escape the integral. Goodbye denominators! To Abbadon with you!
INT([p^N*(1-p)^M]*[p^L*(1-p)^R],p,0,1)/[M!*N!*R!*L!/(M+N+1)!(R+L+1)!]
Rearrange a little bit, combine like terms...
INT(p^(N+L)*(1-p)^(M+R),p,0,1)/[M!*N!*R!*L!/(M+N+1)!(R+L+1)!]
And now it's the same form Keltham had again! She can use exactly what she used to normalize them earlier! She doesn't even have to do any work! She feels like cackling. She doesn't of course, but she'll remember this later and cackle.
[(N+L)!(M+R)!/(N+L+M+R+1)!] / [M!*N!*R!*L!/(M+N+1)!(R+L+1)!]
Clean this up...
[(N+L)!(M+R)!(M+N+1)!(R+L+1)!] / [(N+L+M+R+1)!*M!*N!*R!*L!]
So this could be an answer. But calm down. Don't get overexcited.
OK, that's impossible.
But she still had to be a little careful here. First, was she supposed to normalize? She thinks how the flat probability distributions would've looked. P(p) = 1 from 0 to 1 would be the flat one, that normalizes properly, she knows. If she integrated that times itself, obviously she'd get 1. Concerning. So she has some normalizing work to do still then.
What if it was P(p) = 2 from 0 to 0.5, for each? Then she'd get 2, from 2^2=4, then 4*0.5=2. Makes sense, they're twice as much like each other. So the idea is at least relatively correct, good, good. Think of her answer as a Rating for now, rather than true probability.
Rating = [(N+L)!(M+R)!(M+N+1)!(R+L+1)!] / [(N+L+M+R+1)!*M!*N!*R!*L!]
It occurs to her now that if you assume each true probability can be anything between 0 and 1, the chances they line up exactly should be 0. In a way, it's nonsense to say they can be "the same", at least when working in this framework.
But they can still be nearer together or farther apart. Maybe what she's looking for is the expected difference in probability, or something like that. The half-full ones were twice as good. And clearly, the half full ones are twice as near together. So distance apart is inversely dependent on rating, almost surely.
What's the average distance apart for rating 1 then? That's the key to all this, she can work from that to get everything. But there's something tricky here, she feels a tinge of suspicion.
Owl's Wisdom.
And she realizes she's at least a little wrong. The 0.5 and 0.6 spikes she thought about before would have rating 0, and they're very definitely 0.1 apart. Darn. Is this the end of the road for the two-function method? But this sort of thing would have to happen no matter how she does it, wouldn't it? If she's working from an initial prior that the true p1 can be anything between 0 and 1, and the true p2 can be as well, then the odds of them ever being the same must always be zero.
She's so tempted to ask Keltham what the heck this problem is even supposed to be then, but she forces herself not to. This must be part of the test.
So let's think about this a little more, in a new and strange direction. Imagine she had a prior not about p1 or p2 individually, but about them being the same. If her prior was 1/2 say, that's sorta like saying p1 or p2 can each be one of two values with equal chance, and they might line up or might not. And 1/3 chance of being the same would be three different values, and so on.
So maybe the chance p1=p2 is something you have to get both from the prior probability "Q" that p1=p2 and from the data itself. Maybe her rating is useful after all? Let's think of some cases.
0.5 spike and 0.6 spike. Rating zero. Chance they're the same, always zero. Full-flat and Full-flat. Rating 1. Chance they're the same? Well, the full-flats are like having no data at all, which means the chance must remain Q, the prior you started with. Half-flat and Half-flat. Rating 2. Well, you definitely don't multiply, since Q might be more than 0.5, and 2Q would then be more than 1. Bad. But the chance is definitely more than Q. 0.5 spike and 0.5 spike. Rating... rating infinity. Has to be, infinitely squished together so infinitely big rating. Probability has to be 1.
So what sorta function looks like that? Multiplying is dumb, what about an exponent? But Q is less than 1 and big Rating is good, so maybe...
Updated Probability of Same = Q^(1/Rating) ???
It's a wild guess, but it gets points for being a simple guess, at least. So this would mean half-flats with rating 2 would give you SQRT(Q). 0.5 upgrading to 0.707. It seems plausible? It's at least something to use as a backup plan, it's not a terrible try.
Can she work it out from first principles now that she has a better idea what she's doing? Or maybe find that it's wrong and see something better? Owl's Wisdom runs out, and she decides to cool off for a little, do some sanity checks on her Rating to make sure it even makes sense. It gets better and better as both cluster to the same side, and worse and worse as they cluster to opposite sides. OK, good.
She's pretty sure now that the problem needs a Q. The way it's framed doesn't make any sense without it. If there were buckets, you could make guesses about bucket priors and it might be doable without a Q, but there are no buckets, and buckets are mean and nasty anyway. They went over that. And if you take a totally random 0 to 1 as the prior for both, then the answer to the question is just zero, and it's boring. You need a Q.
But how do you go from Q to anything useful? As necessary as it is, it's kind of an obnoxious object. She thinks about it hard, doesn't get anywhere, and then decides it's time for Fox's Cunning.
Let's go back to those half-flat functions: 2 from p=0 to p=0.5, 0 elsewhere. Imagine I'm given Q=0.5, and that function for each data set. What do I conclude?
It's difficult because the probability weight of the functions together is sort of fundamentally a line shaped-thing, and the functions apart is an area shaped-thing. But she knows it isn't an infinite update in favor of them being not the same, that'd be silly. The two full-flats make for no update at all. Maybe she can use that? With the full flats, the Rating is 1, and Q isn't updated at all. That means Same and Not-Same had the same likelihood there. For the half flats, the Rating is 2, so in a sense the likelihood of Same has doubled. The likelihood of Not-Same... maybe that can't really change? The total probability area can't really be effected by the little probability line.
So imagine a Rating of 2 is a 2:1 update. That feels right, in a comforting way. Her ratings are just likelihoods, basically. So...
Updated Probability = 0.5*2/(0.5*2+0.5) = 2/3
Great, that makes sense. Or generally...
Updated Probability of Same = Q*Rating/(Q*Rating+(1-Q)*1)
Yes, yes. She's going to register a second instance of cackling to be saved for later now. So in summary...
Prior Probability of Same = QSet 1: L Lefts, R RightsSet 2: N Lefts, M Rights
(N+L)!(M+R)!(M+N+1)!(R+L+1)!Rating = -------------------------------------------- = Likelihood Ratio of 'Same' vs 'Not Same' (N+L+M+R+1)!M!N!R!L!
Updated Probability of Same = Q*Rating/(Q*Rating+1-Q)
In Civilization, they wouldn't even give a prior Q as a guess in most cases, just the "Rating", aka the Likelihood ratio. But she figures it's good to be very clear about how one should handle their Q, if they had one and wanted to do something with it.
Do her old sanity checks still work? Rating of zero gives update to zero. Rating of infinity gives update to one. Rating of one still gives "update" to Q. And this is more like the real language of probability than her first guess, it's actually justified, at least sort of. She thinks that for now this is as good as she can get with the main part of the problem.
But this is so so so important. She gets her second Fox's Cunning from staff, and then spends the first minute of it looking over everything she's done again, just in case. Nothing else catches her eye, so she starts thinking about the rest in truth now. So she starts writing paragraphs after all the equations interspersed with notes:
"Now imagine that the p you're looking for really is the same, and you have some medicore Prior Q of that to start with. Let's say p=0.5. But one experiment is leaning right a little, and one left a little, so their true probabilities are p1=0.45 and p2=0.55."
"With small amounts of data in both experiments, the probability functions won't be very sharp. They'll just be soft hills, taller in the middle, and the Rating you get by combining them will still be higher than 1, and update Q in the correct direction, towards 'Same'. But as you took more and more data, the functions get sharper and sharper, like her imaginary spikes. Eventually you'll have spike-like functions at 0.45 and 0.55, and they'll barely even intersect! The rating would be terrible, much less than 1, and update Q in the wrong direction, towards 'Not Same.'"
"So at first, if the experiments just had a little data, comparing them would suggest the right result. But as they got more and more, the problem would get magnified, and it would eventually start suggesting the wrong result instead. It almost seems like a paradox: more data should be making things better, but it's making things worse."
"It's because even if the data is real data, the probability functions from it are sort of lies, if you think they're referencing the true behavior and not just the data from the experiment. p1 and p2 aren't really quite the same thing, they're both just nearby shadows of the same p. So really, the Law isn't lying to you. It's saying p1 and p2 are probably different, and they are! But that's just about the experiments, not about p itself."
"So when you look at an experiment, you should maybe limit how spiky you let your probability function get for the true p. You need to force it to be at least as wide as the inherent errors in the experiment are big. Exactly how wide you force it to be, and how to do that with Law instead of with handwaving, is probably the subject of another problem."
It seems like a big difficult problem, and she's fresh out of Foxes and Owls. So that's that with that for now.
But as lunch approaches and she looks her paper over, she feels confident. Maybe some of this is wrong, it's possible, but she doesn't think it could be wrong enough to sink her.
Korva Tallandria:Korva's Story
Korva misbudgeted her time last night; her requested books about the history of wizard education had come through, and she, idiot that she is, had spent time reading them, hoping that she could make up the five hours it would probably take her to fully understand the Rule of Succession early tomorrow. If not for the instruction to prep specific spells, she might have tried to pull an all-nighter understanding the math (which she did do earlier this week, before their ostensible day off). But even though she can't actually prep Fox's Cunning or Owl's Wisdom, she didn't want to be the only person who couldn't prep anything, couldn't even prestidigitate her name tag. She managed to cram a little work in during breakfast, after spending more effort than usual prepping her spells, but if anything it made her feel more confused.
She should have realized that this was going to be a test day. Stupid.
She has her notes, but there's no time to do the necessary work to fully understand them. She could try copying the exact math that Pilar and Asmodia did, without understanding which parts are different in this problem, but Keltham will see through that immediately, and - honestly she's not sure she can do the math, actually, she looks at it and tries to read the problem and there's just - nothing, no understanding left in her brain. She tries copying the literal symbols that she wrote down in her notes, in case that jogs something loose, but it doesn't seem to; she keeps staring at it without comprehending.
She's going to die.
No, worse than die; she's going to be rendered completely useless for the entirety of her existence. Maybe she should take some comfort in the fact that this cruelty to her must be making Keltham more evil than he already is, but she doesn't think that's the sort of contribution to evil that gets you a duchy, or a spot in the library of oaths, or a few more years of mortal existence. She hates Keltham, violently and passionately, for stealing her perfectly good life from her and buying her like a slave, without so much as checking beforehand whether he would want to use her or cast her aside like so much refuse. He is able to do so, and so he did, and it doesn't really make any sense to be angry about people using the power they have to hurt you however they want, but she hates him for it anyway. She wants to punch him, or maybe cut his head off, or maybe rip his eyes out, or maybe just scream at all of the gods who are so obsessed with him that he's a self-important teenage idiot, that he's nothing special among his people and only very coincidentally of any interest to theirs, that he's a cleric (a fourth circle cleric!) of Abadar, who is nonetheless too careless to realize that he owns his trading partners like chattel, and that any talk of paying them is farcical, and that in general he deserves to be in the same stupid, awful position as the rest of them, and not only doesn't have the basic residual humanity to feel embarrassed about his wholly undeserved position, he doesn't even notice that it's undeserved. Hasn't even noticed that he's an incredibly shitty teacher, even though he's been trying to cram months of lessons into them in a matter of weeks, because he ASSUMES that adults can learn things that much faster than children, even though they CAN'T, there's NO reason to think they could, they can't do it with languages and they PROBABLY CAN'T DO IT WITH MATH EITHER, unless they are GENIUSES who are doing WAY MORE WORK THAN THEY SHOULD HAVE TO DO to make up for his SHITTY ASS TECHNIQUE.
Except that some people can. The other people in this class can.
The only person she hates more than Keltham is herself. Her hatred for herself is blinding, searing, like being shot through with a piece of the sun. She thinks that she might feel a small piece of what Asmodeus feels about her, some tiny percentage of the full weight of the agony of being human, and a not even very impressive human at that.
The only thing she can think to do at all is to write something. Not math; she doesn't have any math in her. She has words, ugly and shifting and imprecise and disgusting and human, but Keltham prefers wrong answers to no answers, so she will at least try to give him the stunningly wrong answer that she has.
There are two tables, each of which produces sets of RIGHT and LEFT. She wants to know whether they output the same or different frequencies of LEFTs and RIGHTs. The first thing to note is that, per previous lessons, you can't know; a Civilization experimentalist would, what - they would have multiple possible specific underlying frequencies of LEFT and RIGHT for each source, to be hypotheses, and then calculate the likelihood of getting each specific dataset in each of the worlds where a given possible frequency was the true frequency. And then they would - what - you need to compare them, but you're not supposed to use hypotheses that rely on comparisons between multiple data sets -
That also doesn't use the Rule of Succession, obviously, because she's only dimly aware of what the Rule of Succession even is. It's something to do with breaking the space of possibilities into infinite pieces, which you'd think would allow you to identify the specific hypothesis that is most likely, maybe, for each of the datasets. And then - how would you present it. The most likely frequency, plus how likely this dataset is if you have the true frequency - maybe, although that has as much to do with the size of the dataset as with whether you're right -
And then given that information, you - you - well, you can at least say how likely it is that the data sets have the same frequency. And then you'd have to... calculate it. Somehow.
Her paper is wet.
She looks at the calculations in her notes, one last time, and her brain just slides off of them. Might as well be written in Celestial or ancient Azlanti.
She writes her stupid, probably-incoherent verbal description of how a Civilization experimentalist might structure their analysis, doesn't do any actual calculations, buries her head in her arms on her desk, and waits for someone who thinks that the garbage never cries in Taldor to kill her and replace her with someone less distraught. One of the other students will target her later anyway, even if security doesn't.
Alexandre Esquerra:Alexandre's Story
So the simple solution is that he lists a set of possible starting theories, doing his limited best to carve up all available regions of idea-space visible to him, then describes which sequences he would consider evidence for each theory, then describe how to compare the relative ratios; again, this is all simple things Keltham taught them. He'll also want to come up with a way to mathematically describe 'all theories have unexpectedly low predictive power'; that ought to be very simple given that Keltham already told them what it looks like...
... The obvious way to note an error is just them getting very different results. In particular, getting results that form very different curves; if it looks like slightly different concentrations along a curve when you combine them, or one of them being thrown by one odd result, those are checkable things, but if, in general, it looks like the thing they are measuring has a different shape, probably the thing they are measuring has a very different shape; if they can’t predict each other, a very plausible outcome is that they aren’t the each other they’re trying to predict, he can put that into math...
And Alexandre will quietly enjoy himself, producing not really a general solution for the problem, because given that individual people will start with their own individual beliefs, many of which they are not explicitly writing down anywhere, he does not think he can do that, but something that, given a little work to turn his neat little descriptions into pseudocode, would work quite well as a computer program for interpreting results and adjusting your priors into posteriors based on them, if any computers happened to exist in Golarion that could run it.
dath ilan:Dath ilan commentary
...Alexandre's methodology would work great if Golarion had a hypercomputer that could check an infinite variety of infinitary metahypotheses, yes.
It's getting anything done with a finite amount of computation that makes SCIENCE!analysis be nontrivial.
But it sure beats not even knowing how to solve the problem using a hypercomputer, and frankly, not all of the researcher-candidates are getting anywhere near that far.
Carissa Sevar:Carissa's Story
Carissa should have put these students through more stressful things so they wouldn't be so stressed out by tests. That's a failure of Chelish education having failed to give students something to be more scared of than math -
- unless the problem is that the students are scared of Hell, and therefore not solvable that way? But they're not even sending the rejects to Hell!
She entertains the temptation to order everyone to believe that they're not going to be executed for math inadequacy, just so that when this situation inevitably happens anyway they'll know it's their own fault for disobeying orders. Somehow, though, it doesn't seem like that would actually fix the problem.
Carissa has a headband that amounts to Fox's Cunning all the time, and has taken to preparing three or four Owl's Wisdoms a day; it's an unfair advantage but life, and death, and everything else, are unfair. Keltham wants a rule for calculating a likelihood ratio between the 'same' theory and the 'not same' theory.
If they're in fact the same, then she can add up the lefts from both experiments and the rights from both experiments and get odds of left:right of (N1 + N2 + 1):(M1 + M2 +1). If they're different, then the odds for the first experiment are (N1 + 1):(M1 + 1) and the odds for the second experiment are (N2 + 1):(M2 + 1).
Then to get the propensity p from the left:right ratio you have to do that deeply unpleasant thing with the factorials, which is going to make everything else messy. She digs it out of her notes.
p: M!*N!/(M+N+1)!
p1: M1!N1!/(M1 + N1 + 1)!
p2: M2!N2!/(M2 + N2 + 1)!
pcombo: (M1 + M2)! (N1 + N2)!/ (M1 + M2 + N1 + N1 +1!)
Right, okay, so the odds ratio between the SAME and NOT-SAME theories is just pcombo/p1p2. Which looks hideous, (N1+N2)!(M1+M2)!(M1+N1+1)!(M2+N2+1)!/ (N1+N2+M1+M2+1)!M1!N1!M2!N2!, but conceptually it's not all that hideous.
Presumably if you'd done this experiment you'd want to report the actual data, and the counts of 'left' and 'right', and also your likelihood functions on the actual propensities, and then your likelihood function on SAME: DIFFERENT.
She feels like the third question is getting at something particular, not just as the general principle that if your two data sources are different then you wouldn't want to combine them and treat them as more observations from the same process, but she's not actually sure what in particular. Presumably you could just notice that the dense bits of your likelihood function for each look very different from the dense bits of your likelihood function for the combination, and that if you ever notice that it's a bad sign. - are they supposed to formalize that? It seems surprisingly hard to formalize, but she does have almost two hours to go....
Some of the new students look miserable. To be fair she's not at all sure that without a headband she wouldn't be just as miserable. She's going to - not try to formalize that, she thinks, and instead fold up her notes and hand them in to Keltham and then tilt her head at him invitingly and head to the library.
dath ilan:A median dath ilani's Story
The median dath ilani has - by Golarion's standards - some power that Golarion knows not, or aptitude that it knows very little. It isn't easily captured by the INT score that is measured by Detect Thoughts. Even if you add in whatever "Wisdom" is measured by Detect Anxieties, there's some residual that isn't measured still.
It's not just about the training the dath ilani undergo in childhood. No, really, it's not. They have been doing a lot of mate-selection, and not just since there've been prediction markets about what kind of children would result.
If somehow you took the median dath ilani, and exposed them to only a few fragments of Law such as Keltham has already taught, a few stories about cognitive science such as Keltham has already told, if they'd had only that little true education by the time they'd reached adulthood -
- the median dath ilani would still be told of the Law of Succession, and think spontaneously, without any outside prompting, of how you could use that to guess whether two sources were the same or different. Just look at the scores if you use the Law of Succession twice, separately, versus using the Law of Succession on both of them together. The difference between those scores is the likelihood ratio for different versus same.
If you get 4 LEFT and 1 RIGHT off one source, and 4 RIGHT and 1 LEFT off another source, then analyzing them separately gives you scores for each of 4!*1!/6! = 1!*4!/6! = 24/720 = obviously 1/30 just look at the factorials. Analyzing both datasets together gives you 5!*5!/11!, which you can see intuitively is going to end up around 1/1024, minus over a bit for the product being 1/2 * 1/3 * 2/4 * 2/5 * 3/6 * 3/7... instead of just 1/2^(10). (It's actually 1/2772 if you bother to calculate - not as nightmarish as it looks, you can cancel a lot of factors.)
Point is, you've got a set of two likelihood functions for two separate datasets, versus one likelihood function on the combined dataset, where, if you started with a uniform prior on three propensity spaces, and multiplied by all those likelihood functions, the two separate functions would destroy all but 1/30 of the probability on each of the two separate spaces, and the combined likelihood function would destroy all but 1/2772 of the uniform probability on its own parameter space.
It's not to the point where if you literally pulled a coin and flipped it 5 times twice, you'd conclude shenanigans from seeing 4 LEFT and 1 RIGHT the first time, versus 1 RIGHT and 4 LEFT the second time. That's just a likelihood ratio of 1/900 vs. 1/1024.
But if the problem is more mysterious than that? If you are less certain at the start that your data is coming from a single source across both cases? Then you'd be looking at an update of more like 2772:900 ~ 3:1 that they were two separate sources, after that. If it wasn't pretty plausible to start, it's not plausible yet now, after so little data. If you were already pretty suspicious, you're now quite noticeably more suspicious, though.
The median dath ilani - even given only such education as Keltham has already provided - fewer hints than that, even - would spontaneously generalize the principle of taking alarm if two experiments seemed to have nonoverlapping likelihood functions.
Suppose the likelihood functions are over a simple hypothesis space - such that likelihood functions form clouds naturally visible in that space - such that there is a natural way to informally see boundaries around narrow subvolumes of the clouds.
You can't say it in an absolute way, apart from some prior and arbitrary concept of how to draw boundaries like that and divide up the space. You could always throw some random points into an otherwise compact cloud and say that you thought they should be in there.
But informally, it's natural enough to see 376 LEFTs and 624 RIGHTs, look at the likelihood function P(data|propensity=p) = p^376*(1-p)^624, and say, "That cloud has 90% of its density between p=35% and p=40%."
If you widen to the amount between p=30% and p=45%, that's 99.9999% of the likelihood density.
And then let's say that you run a different experiment, and it turns up 602 LEFTs and 398 RIGHTs.
Informally - for there is no way to say it formally, without introducing an arbitrary note of subjectivity; we are looking to cues that the data gives us to look outside our hypothesis space, and there is a limit to how much you can ever formalize that, without invoking enough Law to create a mortal from scratch - informally, you look at that and say "No way in superheated toilet paper are those the same two data sources". The overlap of the two clouds in likelihood-space is virtually zero. They have each eliminated practically all of the probability from any hypothesis that could non-stupidly account for the other, and those two different stories cannot exist in the same world.
You do not "combine the data from the two experiments to estimate the parameter". No, not even in sane worlds where all you do to combine the analyses from two experiments is just multiply their two likelihood functions together. Sure, you can multiply p^376*(1-p)^624 by p^602*(1-p)^398 and get p^978*(1-p)^1022, but you obviously shouldn't do that. The first trial concentrates 99.9999% of its survivability between propensities of 0.30 and 0.45, and the second concentrates 99.9995% of its survivability between 0.53 and 0.67. There's no overlap between what the two datasets say are the livable regions of the parameter space.
These two experiments were not conducted with the same world feeding them their answers, though, ultimately, they were conducted inside the same greater Reality. Something is wrong in one place or both.
It's just one of those things that, like, is incredibly important to doing real Science! and jumps directly out at you, once you start thinking about experimental reports in terms of likelihoods, which, in fact, the median dath ilani will do even if nobody explicitly tells them so and even if their entire world tries to tell them otherwise.
Asmodia:Asmodia's Story
Did Asmodia always have that strange thing in her, that the median dath ilani possesses? Did she have it before Otolmens touched her? Did she have it before her hour-long epiphany wearing an artifact headband?
Whatever that strange quality is, Asmodia has it now, somehow, outside of dath ilan.
And if she is not a dath ilani yet - not any kind of ilani, for being so untrained - she is now recognizably an untrained ilani.
Point being, Asmodia already thought through all of Keltham's questions earlier, during the Law of Succession lecture. He'd certainly hinted heavily enough that he was going to ask questions in those directions. Practically spelled it out, even.
Asmodia writes down all the obvious stuff. Thinks a bit more.
Writes down some speculations about, well, maybe you could also update the Rule of Succession on the first data-bunch, ask how well it expected to score, and then further-update the Rule on the second data-bunch to see if it scored around that well; and also do the reverse; and if in either case the further-updated Rule scored much more poorly than it expected to score after the first update, that might indicate a problem.
But it's not really a three-hour question? Is she missing something? If she's not missing something, Asmodia is sort of worried about how much Keltham is softballing the new researchers, here. This seems like really basic and obvious Law of Probability. What's the catch?
She spends a while staring at her paper, but when Carissa Sevar hands in her own version, Asmodia sort of quietly sighs to herself, and hands in her own as Keltham walks out with Sevar.
Security: ...Security notifies the Chosen of Asmodeus that Korva Tallandria seems to have broken down in tears in the middle of the classroom, thankfully after the Chosen left with Keltham. Is this happening inside alterCheliax? If it happened in alterCheliax they'd - probably tell Keltham, right? And if it's not happening in alterCheliax, everybody in the classroom needs to be told it hasn't.
Carissa Sevar: - huh.
It doesn't seem inherently implausible that a student in Taldor would start crying. Some of the kidnapped Taldane students cry. However, Keltham will probably go talk to Tallandria, if he learns this, and that seems like a situation where a slip-up might happen. Presumably she's being mind-read? Is she in any state to talk with Keltham if he shows up wanting to talk to her?
Security: Tallandria is currently experiencing an amount of self-hatred that seems appropriate for somebody as pathetic as herself? She's currently imagining being shot through with pieces of the sun until only the tiny fraction of her that isn't pathetic remains, and feeling out whether she could learn to be okay with being a paving stone.
Given that she's crying in the first place, in front of the other students, and that this is incredibly incredibly bad for her self-interest, Security is concerned that Tallandria wouldn't be able to run Bluff on Keltham even though her soul depends on it.
Dominate Person? Toss her to Subirachs for attempted reforging? They also haven't tried flogging Tallandria until her morale improves, which would be the first resort anywhere sensible if they're not pretending to be fucking Taldor.
Carissa Sevar: How about they don't tell Keltham about this, and tell the other students it's not the case in alter-Cheliax, and pull her out an hour before time's up to see if she can be put back into condition to Bluff Keltham.
Also they should loop in Asmodia, who she thinks liked Tallandria.
Asmodia: ...should Asmodia possibly be going back in and telling Tallandria that she's at least got a place working on Asmodia's Wall, if Keltham doesn't want her? Tallandria was pretty helpful about analyzing Nobility Equilibria after she was called in.
Asmodia is mostly worried because she doesn't know if she'd be sabotaging Tallandria even further by telling her that, which is why Asmodia hasn't said it already. Does Sevar know how people work?
Carissa Sevar: Well, it seems hard to sabotage the girl any harder, plus even if she pulls it together and figures out the math they probably don't want to allow someone with composure problems to have Keltham-contact anyway, so Asmodia might as well go in and say it.
Asmodia: Right. Asmodia will go write down "If Keltham doesn't hire you, I'm planning to assign you to work on my Wall" on a piece of paper and hand that to Korva. She's not sure if Korva is in shape to hear a Message that's spoken once and can't be reread.
Korva Tallandria: Korva lifts her head up and reads the paper.
It makes sense. She's shit at math, evidently, but she dimly remembers that she also gave herself even odds of failing out last night, and she was planning to perfectly calmly angle for a support position of some kind - maybe writing out Keltham's lessons in a form that somewhat dimmer people have a prayer of understanding, once she's put more hours in and figured out what's going on with them herself. But the shadows that real things cast on other pieces of reality - she's good at that. And she'll learn things, which is the sort of thing that might make her soul non-worthless again.
The biggest immediate problem now, then, is the crying. Which - well, from a wall perspective it's not a problem? She knows, on about two seconds' thought, exactly why her alter-self is crying, and she could probably bluff Keltham dead, about this particular thing. Which is convenient, because she still feels much nearer dead than she'd like.
Security: ...Security will relay to the Chosen that Tallandria seems much more together immediately after reading Asmodia's note.
(Possibly the newbies still don't, like, particularly believe Sevar about some things? Maybe it's time to start using mind-control on them, Security does not see how Sevar could have made herself any clearer.)
Tallandria is thinking particularly that she could bluff Keltham about why she's crying. Tallandria's thoughts went immediately to particular stories that seem reasonable to Security, and about how her general composure will seem more consistent later with her having had a breakdown now. Possibly relevant if they want to tell Keltham now that Tallandria's broken down, and... tell him one less lie, he supposes?
(This Security has never had an easy time intuiting the exact outlines of Sevar's 'minimize lies' policy, and is just throwing everything to her.)
Carissa Sevar: Conspiracies are probably more likely to cover up crying breakdowns which they could easily cover up than to not do that, so it earns them some points, if Tallandria's bluff is really good enough, especially in light of the fact that the Taldor girls do sometimes have crying breakdowns.
Hit her with a splendor, just in case, and then tell Keltham.
....she should ask Subirachs about the idea of using Suggestion to make all her underlings believe she's not going to have them permanently reduced to rubble for no reason. It's very appealing but maybe there's some reason this isn't standard which isn't just the cost of the Suggestions.
dath ilan: Among the few things that dath ilan has in common with Cheliax, is that it takes a fair amount to make an adult dath ilani break down in tears in public. Math tests won't do it, even math tests with their future careers at stake.
When Keltham allowed himself to cry about how a hundred and fifty million people weren't going to die for his having entered their world, he sent Carissa out of the room first. Not from thinking that he was doing something bad that must be hidden - Keltham did not try to hide from her afterwards that it had happened. But their relationship being so new, he wasn't sure how the alien might emotionally brain-update about him if she witnessed it directly; and that anxiety played into a different social convention out of dath ilan, that he and Carissa hadn't yet had the conversation that you have before you cry in front of someone, or show other signs of great emotional distress.
Dath ilani do not break down and weep in front of strangers. It's not that they hide weakness. It's that you don't lay that on someone who hasn't agreed to do emotional labor about you.
If it happens anyways, something is really wrong, wrong on a scale that transcends bothering all of the people around you.
Keltham: Keltham will listen to the Security notification with a deer-in-glarelamp expression, and then turn to ask Carissa for advice. In particular, should they possibly go get Maillol. Maillol seems like he might know how to handle this situation.
Sure, as Founder, this is his responsibility, possibly even his fault. It doesn't mean Keltham actually knows how to handle this situation worth a noodle.
Carissa Sevar: " - I don't know that you have to handle it? Maillol has experience commanding teenagers, so I guess he'd be the person to ask if you want to ask somebody, but the emotional wellbeing of every person here isn't your responsibility. In dath ilani companies if an employee of yours started crying would that be your job to solve?"
Keltham: He's the Founder. Everything is his job to, at the very least, make sure somebody is solving. If there's no one person whose job it is, then it's Keltham's job.
Carissa Sevar: " - all right. Let's go ask Maillol. Emotional support and guidance is normally a priest sort of job anyway."
Keltham: Keltham's reflexes out of Civilization do not particularly call for him to walk places to talk to emergency responders during mental health emergencies! That's what phones or Security are for! He'll start striding quickly towards the classroom while asking Security to have Maillol meet them there. Is the situation in the classroom currently stable?
Security: ...yes, Asmodia happened to be nearby and took temporary responsibility for Tallandria.
Keltham: (...he would've guessed Ione, failover to Pilar, especially since Asmodia had already handed in her work.
Slightly confusing but not obviously important, possibly indicates something along the lines of 'Asmodia asked Security to keep a lookout for anything she could impressively assume responsibility for handling'. No obvious Conspiracy correlates.
Keltham is noticing it simply because he is trained to notice confusion.)