Inner Peace (Through Superior Firepower) – Episode 030

[BLANDCorporatio]

BLAND's calculations showed that, in a purely random distribution, FAQ's lack of duplicates would occur about 1 time in 8 (0.12 => 12%), which is well within plausibility (as the 3-heads-in-a-row example shows). How do you go from that to excluding the random hypothesis completely?

Minor quibble here- I computed assuming Faq had 9 casters, for some reason, and that result is with Thinkas twice as likely as any other caster type. It doesn't hurt the question you're making, since having 8 no-rep casters, when all casters are equally likely, will improve those odds; just wanted to make the thing clear as to where the 0.12 comes from. (For all casters equally likely, no rep in 9 tries was about 0.17).

[Kreistor]

BLAND, how does your math change if we assume it's "impossible" (would require Titantic interference, but otherwise won't happen) for a side to naturally pop a duplicate caster (that is, a caster they presently have, not one which they had but croaked). What are the odds then for an 8-caster side lacking a Thinka?

I think Bland missed your Q.

Okay, there's a complexity here that is incredibly difficult to deal with. We can't duplicate casters on pops, but we can on captures and hires. So, I am going to assume that captured duplicates are not Turned, and Hires are never duplicates.

This leaves us with a number of possible combinations of 23 caster types and 8 casters at:

Total = 23!/15!

The number of combinations that exclude Thinkamancers is:

Non-think = 22!/14!

Odds = (22!/14!)/(23!/15!)
=15/23
= 65%

There's a 65% chance that an Side with 8 unique casters that intentionally seeks non-duplicates does not include a Thinkamancer.

[Balerion]

You can't look at the 17% chance of Faq being this way in isolation; we have to observe the chance that every side we had seen so far would have no duplicate casters. So not only is it 17% that Faq would have no duplicates, it was then 50%ish that gk would have no duplicates, a similar number for Unaroyal, the same number for Jetstone... you end up with something closer to 5% that we would observe the current world with the pure random model.

This is not sufficient to reject the model conclusively; ideally we want to push it past a 1% chance. However, it makes assuming that this is the functional model a bad assumption. The math Bland kindly did showed that in around 95% of the cases, that pure random model is wrong.

Now, we have a terrible sample set, so if we actually figured out our variance on that % it is going to be something huge. But that doesn't change what the smart assumption is based on the data we have

[drachefly]

If I get your argument right, drachefly, you're saying that the various alternative hypotheses about caster popping are less information costly because of Signamancy and similar effects (possibly) dictating or indicating something deeper that controls not just caster selection, but various other aspects of a side. ... (ie, while it looks like random u.i.d. deserves the bigger slice of the pie, it's actually not the case).

They are less information costly than they would be if we didn't have this background information and were basing things off the real world. If, for instance, signamancy was not known or largely not understood (e.g. we were discussing this before the discussion of signamancy in book 2), the hypothesis that the (non-magical) makeup and clothing that a ruler wears could have bearing on whether casters pop and what type they are would be horrendously unlikely, simply from how much detail you'd have to include to specify what you were talking about.

Essentially, the possibility of such a connection has been brought to our awareness at the start of this investigation, through other things we know about Erfworld.

With all of the factors that could reasonably be contributing, and all of the combinations of them that could be applying, even before we begin measuring evidence specific to this case (sets of casters on each side), though the null hypothesis still has the largest individual probability of any of the rules, it is already a minority of overall weight of the hypotheses.

A bit like with a gas - even at room temperature, the most occupied microstate for the air in your room is for it all to be lying on the floor almost motionless, but there are so many states in a 'normal' macrostate, and they are not that much less likely individually, that you're not going to find the air in your room doing that in the projected lifetime of the universe.

We're not in the thermodynamic limit with the casters, obviously.

My question is how can one go about actually assigning numbers to such things, be it in this or some simpler example that may illustrate the process better.

That is the trick, isn't it?

We have some experience with Erf mechanics. Without specifying which one we're talking about, how likely is it that something out of conscious control... involves natural signamancy? Natural thinkamancy? Fate? Non-independent, biased towards anticorrelation (like drawing cards)? Biased towards correlation? Is terrain-based? is based on something nothing else is based on?

If we had enough such mechanics we could estimate the chances of it involving multiple different factors. For now, independent probabilities seem a reasonable starting point. Thus, the prior probability of our u.i.d. 'null hypothesis' will be the product of the probabilities of each of those not being the case.

Putting numbers to it...

the u.i.d null is: "luckamancy is the only thing at work determining this, and the probabilities are even"

Signamancy, luckamancy and fate are pervasive in these kinds of effects concerning how things out of your control work out, but fate could be overrepresented due to our seeing the narrative. Let's call it 1/5 on signamancy, 1/10 on fate. Let's drill down into luckamancy with correlations and anticorrelations (if it's neutral, that's the u.i.d null right there)
Anticorrellations - luckamancy involve shuffling numbers around in a way that seems rather more like a deck of cards than dice. If that's a global effect in some way, it would naturally work out non-u.i.d. even without it being a mechanic specific to this subject. Let's call it 1/6.
Correlations... well, luckamancy could go the other way too, but we haven't seen it. Let's call it 1/200.

natural thinkamancy - what you really want, subconsciously... we haven't seen this at work, but there's an obvious mechanism in related matters like overlord's awareness of side status, build orders, etc. Let's say 1/5.

terrain - unit types poppable totally depend where you are. I would put that really high, somewhere around 1/4
Novel, specific factors - we've run into new things behind the scenes many times - around 1/6.

So... the prior probability of u.i.d. is, by this measure: 4/5 * 9/10 * 5/6 * 199/200 * 4/5 * 3/4 * 5/6 = 30%

[BLANDCorporatio]

BLAND, how does your math change if we assume it's "impossible" (would require Titantic interference, but otherwise won't happen) for a side to naturally pop a duplicate caster (that is, a caster they presently have, not one which they had but croaked). What are the odds then for an 8-caster side lacking a Thinka?

I think Bland missed your Q.

No I didn't, and for once our answers match.

Spoiler: show

Kreistor's answer is more compact, because he uses as intuitively obvious the notion that "the probability of a collection of casters is not dependent on the order in which those casters were popped".

A feature/bug in my tackling/presenting arguments that involve counting odds is to avoid shortcuts, as I don't trust them. But they often do work. For example: suppose we are given a set of 8 distinct casters selected from 23. 8! (aka, all permutations of 8) is the number of possible pop histories (assume no deaths or emmigration or whatever other loss). There are arr(23, 8) possible pop histories (whether consistent with our caster selection or not). So the probability of the sequences that are consistent with our caster selection (we assume all equally likely, independent sequences) is 8!/arr(23, 8). But that number turns out to be exactly 1/com(23, 8) where com(23, 8) is the number of possible selections of 8 elements from 23 (and we ignore the order).

To tackle 0beron's question, one may reason in yet a third, even shorter way. Basically, you get a chance to pop a Thinkamancer on eight pop occasions. You get a thinka if one of those chances yields one, and they are all mutually exclusive so you just add them. First chance is 1/23. Second chance is (1-1/23)*1/22 which, amazingly, is 1/23 again. If the chance to pop a thinka on the kth try (and never in a pop before) is 1/23, then the chance to pop on kth+1 is (1-k*1/23)*1/(23-k) which again is 1/23 (1-k*1/23 appears because "pop on k" must fail for k from 1 to, uhm, k). So anyway, 8 tries, 1/23 chance to pop a thinka on each try (and exactly that try and never before) gives 8/23 chance to pop a thinka.

Let's call it {some number}

That right there seems to me like the second step of the Underpants Gnomes' plan.

I'd expect that different results would be possible based on the figures placed in the "let's call it" clauses. How different those results are from what you obtain isn't my point (it may well be that one would need to change the numbers wildly to get u.i.d. with 50% prior probability). My point is that rather than settling the issue, they toss it back a step.

[MarbitChow]

(A whole bunch of stuff)

Truly, you have a dizzying intellect.

[SteveMB]

You can't look at the 17% chance of Faq being this way in isolation; we have to observe the chance that every side we had seen so far would have no duplicate casters. So not only is it 17% that Faq would have no duplicates, it was then 50%ish that gk would have no duplicates, a similar number for Unaroyal, the same number for Jetstone... you end up with something closer to 5% that we would observe the current world with the pure random model.

Another point: we're mostly seeing the casters the sides have now, with only a few cases of casters who popped to one side and moved to another for some reason. One obvious possibility: a side that pops another caster of a type it already has might trade one of them for a type it lacks.

[MarbitChow]

You can't look at the 17% chance of Faq being this way in isolation; we have to observe the chance that every side we had seen so far would have no duplicate casters. So not only is it 17% that Faq would have no duplicates, it was then 50%ish that gk would have no duplicates, a similar number for Unaroyal, the same number for Jetstone... you end up with something closer to 5% that we would observe the current world with the pure random model.

You can't factor in additional sides and use that to decrease the odds. If you add all sides to the same sample, you now start getting duplicates. If you treat them as independent trials, you only show that the results are even higher than the 17% we've already shown.

Without additional information, we're taught to assume that the simplest explanation is the correct one. We've got 2 simple explanations : it's random, or it's fate. Personally, I'm in favor of the 'it's fate' explanation - sides just get whatever casters they're "supposed to". But we cannot use the evidence at hand to eliminate the 'it's random' explanation. You can't look at a few samples of data and conclude that, because the samples have a somewhat unlikely probability of occurring randomly, that it is unlikely that they occurred randomly. That's a false mental shortcut. There are some patterns that would be so unlikely to occur (like 20 thinkamancer in a row) that you could use that to fairly conclusively disprove randomness, but the examples we've got don't fall into that category.

[BLANDCorporatio]

You can't look at the 17% chance of Faq being this way in isolation; we have to observe the chance that every side we had seen so far would have no duplicate casters. So not only is it 17% that Faq would have no duplicates, it was then 50%ish that gk would have no duplicates, a similar number for Unaroyal, the same number for Jetstone... you end up with something closer to 5% that we would observe the current world with the pure random model.

You can't factor in additional sides and use that to decrease the odds. If you add all sides to the same sample, you now start getting duplicates. If you treat them as independent trials, you only show that the results are even higher than the 17% we've already shown.

What do you mean? Balerion is doing exactly what I recommended done (step number 2) a few posts ago. Basically, yeah, treat all sides as independent trials of the event "no duplicate casters". For some reason, we're assuming this event happened for all known sides, so what we seek is an intersection of (assumed independent) events which is a product of sub-unitary numbers and is bound to be less than any individual number.

[MarbitChow]

What do you mean? Balerion is doing exactly what I recommended done (step number 2) a few posts ago. Basically, yeah, treat all sides as independent trials of the event "no duplicate casters". For some reason, we're assuming this event happened for all known sides, so what we seek is an intersection of (assumed independent) events which is a product of sub-unitary numbers and is bound to be less than any individual number.

But we don't actually even know which casters popped for which sides. We're assuming FAQ and Jetstone popped all their own casters. We know GK didn't - they got Wanda and Jack at least. We've got 1 semi-decent trial (FAQ), and a bunch of aborted / contaminated trials. When there are 23 possibilities, checking the results for only the first 2 or 3 pops doesn't really tell us anything conclusive at all.

[BLANDCorporatio]

But we don't actually even know which casters popped for which sides. We're assuming FAQ and Jetstone popped all their own casters. We know GK didn't - they got Wanda and Jack at least. We've got 1 semi-decent trial (FAQ), and a bunch of aborted / contaminated trials. When there are 23 possibilities, checking the results for only the first 2 or 3 pops doesn't really tell us anything conclusive at all.

That's fair, but different from the previous objection. It's one thing to point out that the trials we know of are problematic because we don't have access to enough exact pop histories, and another to claim that the whole idea of combining several trials (assumed nice enough and independent enough) is wrong. It certainly is the case that, had we several well known examples of pop histories without repetition, the chance of all of them happening under a r.u.i.d. caster type|caster pop would be slimmer than the chance of any single one happening under the same assumption.

[MarbitChow]

That's fair, but different from the previous objection.

Yeah, I still have a hard time with that "show all your work" thing before posting my conclusions.

[drachefly]

Let's call it {some number}

That right there seems to me like the second step of the Underpants Gnomes' plan.

But here's the thing. If you don't talk about it, you're implicitly calling it something. You're implicitly calling it 1 or 0, generally 0.

Calling it 1/10, even if that's a terrible estimate, is far more accurate.

Now, I could have included error bars, and ended up with something like "30% [+48% - 22%]"... but I'm sure you wouldn't be happy with pushing the underpants gnomes to put another concrete step before "???".

Look at it this way. Suppose you're forced to lay odds on this process being u.i.d. and people can take either side of these odds. You are forced to do this knowing all of the other mechanics we've seen in Erfworld, but you don't know anything about the caster distribution itself.

What odds do you offer? Evasion will be taken as concession.

(A whole bunch of stuff)

Truly, you have a dizzying intellect.

He asked for numbers. What am I supposed to do?

[wrecan]

Another factor that could be skewing observations is that the Thinkamancers are the only group of casters that appear to have an overarching organization. It could be that, even if thinkamancers are popped as frequently as other casters, the existence of the GMTA means that thinkamancers have a higher life expectancy in the Magic Kingdom than other casters because the GMTA is able to shift resources to keep their members from croaking.

So our observations may be skewed. We can't assume that the distribution of caster types we observe is in any way correlated to the distribution of caster types that pop.

[drachefly]

Another factor that could be skewing observations is that the Thinkamancers are the only group of casters that appear to have an overarching organization. It could be that, even if thinkamancers are popped as frequently as other casters, the existence of the GMTA means that thinkamancers have a higher life expectancy in the Magic Kingdom than other casters because the GMTA is able to shift resources to keep their members from croaking.

So our observations may be skewed. We can't assume that the distribution of caster types we observe is in any way correlated to the distribution of caster types that pop.

None of the analysis I've seen here deals with casters in the MK, only caster-popping events. My prior on organizational structure impacting pop rates is really really low.

[BLANDCorporatio]

What odds do you offer? Evasion will be taken as concession.

That's a terrible way to justify an answer, but read on.

But here's the thing. If you don't talk about it, you're implicitly calling it something. You're implicitly calling it 1 or 0, generally 0.

Calling it 1/10, even if that's a terrible estimate, is far more accurate.

Here I'd disagree as a matter of nuance. Calling it 0 or 1 is committing to being certain and certainty should remain fixed. So its dangerous to assume certainty, and all the talk about accuracy is imo misleading.

Now, I could have included error bars, and ended up with something like "30% [+48% - 22%]"... but I'm sure you wouldn't be happy with pushing the underpants gnomes to put another concrete step before "???".

Yes I would have, because error bars were not what I was after. You say, calling it 1/10 is more accurate. Let's say, I call it 9/10. What then, apart from (dis)agreeing to disagree about priors?

Look at it this way. Suppose you're forced to lay odds on this process being u.i.d. and people can take either side of these odds. You are forced to do this knowing all of the other mechanics we've seen in Erfworld, but you don't know anything about the caster distribution itself.

What odds do you offer? Evasion will be taken as concession.

So as to not evade your challenge, 0.4158238130346. Natch.

But see that's the problem. You, in your post, assigned various probabilities. I guess the assignments make sense. BUT, they came in the context of your argument with MarbitChow, that we can rule out r.u.i.d. My understanding of the situation is that, were MarbitChow to repeat your exercise, he can find a combination of values that yield 50% or more for r.u.i.d. as opposed to your 30%. So that argument based on priors cannot settle the matter nor imo justify the claim that we can exclude something with (something approaching) certainty.

And to return to something for a moment,

What odds do you offer? Evasion will be taken as concession.

That's a terrible way to justify an answer, but read on.

I can and did offer odds on that. I don't think it follows that I should.

[drachefly]

No. 0.41... was based on the data about caster frequencies. I was specifically asking what you'd say before you knew that.

P(caster frequencies | u.i.d.) = 0.41...
P(u.i.d. | caster frequencies) != 0.41...

You say, calling it 1/10 is more accurate. Let's say, I call it 9/10. What then, apart from (dis)agreeing to disagree about priors?

Actually, if you're talking about Fate, then yeah, 9/10 could be more accurate! There's little that fate hasn't been able to influence one way or another. But whether McFate (as opposed to grand fate) is capable of causing a deviation from u.i.d. is a different question, and I think 1/10 is more likely on that front.

[BLANDCorporatio]

No. 0.41... was based on the data about caster frequencies.

Technically it was based on my fingers mashing the number keys randomly

[drachefly]

No. 0.41... was based on the data about caster frequencies.

Technically it was based on my fingers mashing the number keys randomly

Oh, I see - it was close enough to an earlier estimate that I thought it was the same thing.

Excuse me for not checking carefully.

So, anyway, if you offered me those (essentially) 2:3 odds, I'd totally take the short side of those odds, and in the bizarre case of somehow being able to repeat this bet on independent otherwise identical fictional narratives (ha), I think I would tend to win. 40% just isn't your best estimate.

[BLANDCorporatio]

Actually, if you're talking about Fate, then yeah, 9/10 could be more accurate! There's little that fate hasn't been able to influence one way or another. But whether McFate (as opposed to grand fate) is capable of causing a deviation from u.i.d. is a different question, and I think 1/10 is more likely on that front.

I guess maybe? I mean, in the matter of which Erf mechanic is in place for some particular phenomenon, I'm anumerally agnostic. No number assigned to any various possibility.

I understand the subsequent method- "if you start out believing (aka, assigning these probabilities) x or y, then to be consistent, this is what you must believe after something happens". I'm not sure the previous step, of assigning numbers, is enough as basis for argument.

I mean, we can compute the average of a well known sample, say. It is exactly what it is and there's no argument. OTOH we could go around in circles forever about the role of McFate or FATE and in our little bubble each could very well be right.