# 34

## I.

"Transitivity" is a property in mathematics and logic. Put simply, if something is transitive it means that there's a relationship between things where when x relates to y, and y relates to z, there's the same relationship between x and z. For a more concrete example, think of size. If my car is bigger than my couch, and my couch is bigger than my hat, you know that my car is bigger than my hat.

(Epistemic status: I am not a math major, and if there's a consensus in the comments that I'm using the wrong term or otherwise making math mistakes I can update the post.)

This is a neat property. Lots of things do not have it.

## II.

Consider the following circumstance: Bob is traveling home one night, late enough there isn't anyone else around. Bob sees a shooting star growing unusually bright, until it resolves into a disc-shaped machine with lights around the edges. He finds himself levitated up into the machine, gets poked and prodded by the creatures inside for a while, and then set back down on the road.

Assuming Bob is a rational, rationalist, well-adjusted kind of guy, he now has a problem. Almost nobody in his life is going to believe a word of this.

From Bob's perspective, what happened? He might not be certain aliens are real (maybe he's just had a schizophrenic break, or someone slipped him some interesting drugs in his coffee) but he has to be putting a substantially higher percentage on the idea. Sure, maybe he hallucinated the whole thing, but most of us don't have psychotic breaks on an average day. Break out Bayes.

[WARNING: There's a discussion in the comments suggesting I'm doing the setup and math wrong. Seems plausible, this is my first time doing Bayes with an audience.]

What are Bob's new odds aliens abduct people, given that his experiences? Let's say his prior probability on alien abductions being real was 1%, about one in a hundred. (That's P(A).) He decides the sensitivity of the test - that he actually got abducted, given he had this experience - is 5% since he knows he doesn't have any history of drug use, mental illness, or prankish friends with a lot of spare time and weird senses of humour. (That's P(B|A).) If you had asked him before his abduction what the false positive rate was - that is, how often people think they've been abducted by aliens even though they haven't - he'd say .1%, maybe one in a thousand people have seemingly causeless hallucinations or dedicated pranksters. (That's P(B|¬A).)

P(A|B) = 0.3356, or about 33%.

The whole abduction thing is a major update for Bob towards aliens. If it's not aliens, it's something really weird at least.

Now consider Bob telling Carla, an equally rational, well-adjusted kind of gal with the same prior, about his experience. Bob and Carla are friends; not super close, but they've been running into each other at parties for a few years now.

Carla has to deal with the same odds of mental breakdown or secret drug dosages that Bob does. Lets take lying completely off the table: for some reason, both Carla and Bob can perfectly trust that the other person isn't deliberately lying (maybe there's a magic Zone of Truth effect) so I think this satisfies Aumman's Agreement Theorem. Everything else is a real possibility though. She also has to consider the odds that Bob has a faulty memory or is hallucinating or she's misunderstanding him somehow.

(True story: my undergraduate university had an active Live Action Roleplaying group. For a while, my significant other liked to tell people that our second date was going to watch the zombies chase people around the campus. This was true, in that lots of people looked like they had open wounds, were moaning "Braaaaains," and were chasing after other people. My S.O.'s family correctly did not panic about a zombie apocalypse when they heard we'd watched zombies chase people around the campus despite the literal interpretation of those words being highly concerning. They didn't know what the explanation was, but they assumed the explanation was not the plot of Dawn of the Dead.)

Even if Bob says there's evidence, the aliens did some kind of surgery on him and left a scar the surgical scar isn't ironclad evidence. I have several scars that people don't notice unless I point them out, and I feel like it'd be hard to tell the difference between alien surgery scars, appendectomy scars, and a scar from falling on a nail as a teenager. Plus, and I don't know how to say this politely, but I feel like it's normal to have more confidence in your own mental clarity than that of your friends?

So when Bob tells Carla a story about the aliens, her prior probability is 1% just like his was. (P(A)). The sensitivity of the test - that aliens actually abduct people, given someone is telling her aliens abducted him - is 2.5% since she doesn't really know his drug habits and hasn't ruled out there's a LARP she's missing the context for. (That's P(B|A).) If you had asked her before this conversation what the false positive rate was - that is, how often people tell someone they've been abducted by aliens even if they haven't - she'd say 1%, they don't usually come up to you at nice parties but she can find a dozen in well chosen reddit thread and people at rationalist parties can be kind of weird. (That's P(B|¬A).)

P(A|B) = 0.0246, or about 2.46%.

She makes a much smaller jump, even with our magical cheat above that they can trust each other not to deliberately lie. Obviously if Carla thinks Bob might be lying, then she's updating even less towards aliens. If Carla tries to tell her friend Dean about what Bob reports, Dean is going to update even less. At some point in the chain this fairly convincing experience that Bob had becomes equivalent to "somebody I don't know said they got abducted by aliens." That's already true, and I don't believe in alien abductions right now, so I'm not sure how Dean or even Carla would go about convincing me. That said, Carla at least should be updating a little bit towards aliens existing: conservation of expected evidence, aliens aren't less likely in worlds where your friends tell you they personally got abducted by aliens. (Dean might not update at all; "a friend of a friend says they got abducted by aliens" might already be part of his prior.)

Even if Bob trusts his observations a lot, Carla doesn't trust Bob's report of his observations as much. Trust is a little bit, but not completely, transitive.

## III.

In Decentralized Exclusion Jeff K. suggests that relatively decentralized communities like effective altruism dinners or rationality meetups or contra dances can sometimes exclude people, but that it takes a detailed public accusation. I've noticed that detailed public accusations are a high bar to clear: people worry about excessive reputation damage[1], retaliation, and defamation suits among other things. People disagree about whether the accuser is trustworthy, whether the accusation was shared in the right way, and whether the misdeed is actually as bad as it sounds. What I've seen more of is individual branches of the community internally banning someone, and then news of that ban spreads because organizers talk to each other.

(All specifics in the following example have been changed, but this is a composite of some real examples.)

Say you're one of the main organizers for a rationality meetup in Dallas, and you find out that Toronto's rationality meetup has banned Emma Lastnamehere. Emma then moves to Dallas and shows up at your meetup. There are internal or at least well attested statements from the organizers of Toronto, but nothing big and public, and it sounds like it was a messy and tiring process. You're acquainted with the Toronto organizers, maybe been to a couple of conferences together or hung out on a shared Discord server, but you're not especially close to them. What do you do about Emma attending your meetups?

Maybe your prior on a random new attendee being a problem is about 0.5%. (That's P(A)). What's your sensitivity and false positive rate?

A high sensitivity (P(B|A)) means that if Emma really is a problem, a lot of organizers for groups she's attended will have banned her from their meetups. A low sensitivity means that even if Emma is a problem, it's unusual for her to have been banned by other groups.  It's actually pretty rare for a rationalist group to ban someone, so sensitivity isn't super high. I'm going to call it 30% for this example, but you should think about where your number would be.

A high false positive rate (P(B|¬A)) means that even if Emma isn't a problem, she might have gotten banned from a group or have statements from people complaining about her. A low false positive rate means that if Emma isn't a frustrating or troublesome attendee, it's unlikely anyone bans or complains about her. I think the false positive rate on bans is pretty low; obviously it's hard to tell, but I think getting an initial ban from a rationalist group doesn't happen unless there's been a long pattern of problems. I'll use 1% for this example; there are (hand-waves vigorously) about two hundred rationalist groups and it seems reasonable two or three would ban for reasons I didn't agree with.

P(A|B) = 0.131, or about 13.1%.

Everyone gets to draw their own lines, but I don't think I'd automatically copy a ban from another group. Maybe it was some kind of personal friction, or the organizer was in the wrong. (Though I want to clearly state that I support the local organizer making that call; well kept gardens die by pacifism, and on the margin I suspect rationalist organizers should do more local bans.) I'd keep an eye on Emma, but she can come.

I picked the words "initial ban" intentionally. The first ban is a lot of evidence. What if you find out Emma got banned from Miami as well? The second ban is a decent chunk more evidence, but it depends how independent the bans were. Let's say for the moment they were completely independent. Miami and Toronto never communicated anything about Emma with each other. Now I'm starting from a prior of 13.1%.

P(A|B) = 0.8186, or about 81%.

I no longer want Emma at my meetup.[2] If Emma shows up with an explanation of how the Miami and Toronto organizers are liars and I should disregard everything they say, I'm starting from the prior that she's not a reliable source of information and that accusation doesn't reassure me that I'll be glad to have her around. Trust (and mistrust) does spread through the network, albeit incompletely.

## IV.

There's a real hazard of cascading.

If you told me that someone was banned from six meetup groups, at that point I suspect that groups four through six were probably making their decisions based on the reports from groups one, two, and three. It might be worth talking to them all at least briefly to see where the firsthand evidence is coming from; if group one was the only group Emma previously attended and groups two through six were going off of that first report, maybe I'd give her another chance.

(It's even possible for reports like this to be an example of citogenesis, where the first report wasn't actually a ban but just an incorrect rumour of a ban, the second group banned based off of that rumour, and the first group heard about the second group and actually banned Emma. Check your sources where you can!)

If you already have the evidence that Emma was banned from six meetup groups, how much does hearing a seventh has banned them change whether you'll ban her from your group? Not a lot. Whatever is going on, you already knew she got banned from meetup groups. At some point, the question becomes how much those groups are updating on each other and whether whatever trait those groups have in common is something you share. If the problem is that someone is too loud and therefore has been banned from six different Librarians' Clubs, and you run a Heavy Metal Jam Club, then maybe you don't care about the bans.

If trust is too transitive — if every group mirrors each other's ban lists, if you trust every third-hand report of aliens as much as you trust your most honest friend — then they're really only providing as much evidence as one example. This is one of the central flaws in gossip as a method of conveying someone's reputation. Gossip reliably fails to distinguish between one person experiencing a problem and a dozen people experiencing the problem.

If trust is completely intransitive — if you take no one's word, if you try to figure out everything based solely on your own observations — then you're ignoring genuinely useful signal. This is one of the central flaws in refusing to consider other people's character evaluations. You are going to give Emma the benefit of the doubt, she's going to show up to your meetups, and then you're going to go through a messy and tiring process that you could have skipped.

## V.

I think about Bob and Carla's situation a lot.

The numbers make it feel like a quantitative difference of degrees, when from the inside it feels like a qualitative difference of some different kind of knowledge. Bob remembers the caustic smell of the alien spaceship and the gut-wrenching sensation of being caught in the tractor beam. He's pointing at the star-shaped puckered scar on his hip, pointing out that surgical scars don't look like that dangit. Maybe it's not aliens, fine, but something weird happened and it's forced him to grapple with whether he can trust his own mind.

Meanwhile, Carla is nodding politely and trying to think of how to get out of this conversation. This is a little weird, Bob seemed like a level-headed guy the last time she saw him, but it's not causing her to rethink her place in the universe. Some people are just weird.

I can't think of a way to bridge this. Even assuming Bob and Carla can't lie, and both know the other person can't lie, it's so much more likely that Bob is failing to bring up some reason he would have hallucinated than that alien abductions happen. I've been in Carla's shoes a few times, and even been on Bob's side in much less dramatic or world-reshaping circumstances. Sometimes there's extra proof you can find, but other times you're just left with someone's word.

(And the nagging suspicion that they're lying. But the more maddening circumstances are where the lie just makes no sense. Once, someone sounded deeply certain I had said something I hadn't said, something that would have been very uncharacteristic of me to say, a minute or so after I'd supposedly said it. One of us was wrong, and it was such a weird lie to tell I was confused what the point was of telling it.)

Thing is, if Bob expects people to disbelieve him and think he's nuts if he talks about the alien thing, then he doesn't tell anyone. That would be a really dumb way for aliens to successfully hide from humanity; maybe they abducted half of the world, one at a time, and almost nobody talks about it because we each think nobody would believe us.

But Bob's not wrong! If I had the experience of being abducted by aliens, I don't think anything in my life would go better in the short term from me saying that to people. Long term, it helps everyone else (who can adjust their confidence in me appropriately) and eventually me conditional on this being a mental break that can be fixed by anti-psychotics.

I think there are cases where trust is more transitive. Close knit teams who have worked together for years and shared a lot of models of the world with each other, married couples with solid relationships, your best friend when you both grew up together in a small town and stayed in the town still seeing each other, where if they told you they got abducted by aliens you would actually update as much as if it happened to you. There's one person in my life that, if they told me they got picked up by little green men from mars, I would put higher odds on me hallucinating them saying that than I would on them hallucinating the aliens.

The obvious failure case for the meetup bans is the ban being sufficiently private that other groups never learn about it, leading Emma to get banned from groups one by one after an exhausting process every time. The second, less obvious failure case is Emma getting banned once, a half-dozen other groups mirroring the ban without saying they're just mirroring the first ban, and every other group assuming Emma got banned from seven groups one by one after seven independent decisions. Short term, I don't know if anything gets better for a group that makes a ban public. Long term, it helps everyone else when you're right. It also makes you the target of more focused adversarial action.

I claim trust works best if you say what you think you know, and how you think you know it.

1. ^

If the problem is that someone is a known serial rapist, then excessive reputation damage might be acceptable. (Though you should hopefully be fairly confident about it!) If the problem is that someone is rude and annoying to work with, then a big public statement can feel like overkill.

2. ^

Though now that I've said that, I'm going to have to worry about adversarial action. Expect a less organized post at some point about this, but any time you explain how you make decisions, if anyone whose interests are not aligned with yours would want those decisions to come out differently, you need to be at least a little aware you might be getting set up.

# 34

New Comment

EigenKarma is a (basically abandoned) attempt to apply a semi-transitive trust algorithm in real situations. EigenTrust (the original paper, which is the basis of EigenKarma) had some brief discussions and computational experiments on how adversaries affect trust.

I had been wanting to spend some time working on improving EigenKarma for the last year but haven't got around to do it.

He decides the sensitivity of the test—that aliens actually abduct people, given he experienced aliens abducting him—is 5% since he knows he doesn’t have any history of drug use, mental illness, or prankish friends with a lot of spare time and weird senses of humour. (That’s P(B|A).)

Er, what? P(“aliens actually abduct people, given [Bob] experienced aliens abducting him”) is P(A|B), not P(B|A). This is what you’re trying to find out, so it can’t be one of the inputs.

P(B|A), which is indeed the input you want, would be P(“Bob experiences aliens abducting him, given that aliens actually abduct people”). However, this has nothing whatsoever to do with whether Bob has “any history of drug use, mental illness, or prankish friends with a lot of spare time and weird senses of humour”. (It would presumably depend on things like “what is the selection procedure that the aliens use to choose abduction victims”, plus various figures like “how many people are there”, “what is the probability that Bob would be in a certain location at a certain time”, etc., etc.) And I can’t see where you might get a figure of 5% for this.

So, as far as I can tell, the math you provide doesn’t work.

This part is obviously more speculative, but let’s suppose that Bob reasons thus: given that aliens abduct people, there is perhaps an equal chance that they would abduct any human as any other. (Is this a reasonable assumption? Who knows?) And let’s say that they abduct 100 people per year. So the chance of having been abducted at least once in (let’s say) 40 years of life would be 1 − (7,999,999,900 / 8,000,000,000)^40 = ~0.0000005.

Ah, but there’s a catch! The chance of having an abduction experience if there are aliens isn’t just the chance of being abducted, it’s the chance of being abducted plus the chance of falsely coming to believe you’ve been abducted when in fact you have not. (As surely we do not think that the existence of aliens would prevent humans from having schizophrenic episodes or LSD trips etc.?) Thus we must add, to P(B|A), that 0.001 chance of having a false abduction experience, for a total of 0.0010005. (If we didn’t account for this, we’d end up concluding that Bob’s experience should lead him to revise P(A) drastically down!)

So, the revised calculation:

P(A|B) = (0.0010005 * 0.01) / ((0.0010005 * 0.01) + (0.001 * 0.99)) = 0.000010005 / (0.000010005 + 0.00099) = 0.000010005 / 0.001000005 = ~0.01000495 = 1.000495%.

1.000495%. Not 33%. (The prior, we must recall, was 1%.) In other words, this is an update so tiny as to be insignificant.

Of course we can tweak that by modifying our assumptions about the aliens’ behavior—how often do they abduct people, how do they select abductees—but you’d have to start with some truly implausible assumptions to get the numbers anywhere near a large update.

Bob should notice that it is overwhelmingly more likely that his experience was false than that it was real, and it should have essentially no effect whatsoever on his estimate of the probability of the existence of aliens who sometimes abduct people.

Thank you for checking my math and setup! This is my first time trying Bayes in front of an audience.

Yeah, I think I described P(A|B) when trying to describe the sensitivity, you are right that whether aliens actually abduct people given Bob experienced aliens abducting him is P(A|B). It's possible I need to retract the whole section and example.

Your description of P(B|A) confuses me though. If I think through the standard Bayes mammogram problem, I don't set P(B|A) as P("A specific woman gets a positive test result, given some people get a positive test") and have to figure out what the selection procedure is that the doctor uses to choose people to test. We're looking for P("A specific woman gets a positive test result, given she actually has cancer.") I think Bob gets to start knowing he experienced getting abducted, the same way the woman in the mammogram problem gets to start knowing she got a positive test. He then tries to figure out whether the abduction was aliens or some kind of hallucination, the same way the woman (or her doctor) in the mammogram problem tries to figure out whether the test result is a true positive or a false positive.

Hrm. So, in the mammogram problem, if the sometimes the machine malfunctions in a way that gives a positive result whether or not the woman actually had cancer, then some of the time the woman will coincidentally happen to have cancer when the machine malfunctioned. I think that's just supposed to be counted as part of the probability the woman with cancer gets a positive test, i.e. the sensitivity? Translating back to Bob's circumstances, aliens are real, but Bob hallucinated?

Intuitively it makes sense to me that if someone thinks they got abducted by aliens, it's more likely they're hallucinating than that they actually got abducted by aliens. It's true that aliens actually abducting people wouldn't mean people stop having hallucinations. But adding P(B|¬A) - the rate of false positives - to P(B|A) - the rate of true positives - seems like some kind of weird double counting. What am I misunderstanding here?

Intuitively it makes sense to me that if someone thinks they got abducted by aliens, it’s more likely they’re hallucinating than that they actually got abducted by aliens. It’s true that aliens actually abducting people wouldn’t mean people stop having hallucinations. But adding P(B|¬A) - the rate of false positives—to P(B|A) - the rate of true positives—seems like some kind of weird double counting. What am I misunderstanding here?

Well, to start with, if you don’t include the false positive rate in P(B|A), and work through the numbers, then, as I said, you’ll find that having the abduction experience will drastically lower your probability estimate of aliens. You would have:

P(A|B) = (0.000005 · 0.01) / ((0.000005 · 0.01) + (0.001 · 0.99)) = 0.00000005 / (0.00000005 + 0.00099) = 0.00000005 / 0.00099005 = ~0.0000505025 = 0.00505025%.

So that’s clearly very wrong—even if it doesn’t tell you what the right answer should be.

But, intuitively… well, I explained it already: if aliens exist and abduct people, some people will still take drugs or go crazy or whatever, and hallucinate being abducted. Bob could be one of those people. It’s not double-counting because the A is “aliens exist and abduct people”, not “Bob was abducted by aliens”. (Otherwise P(A) could not possibly have started as high as 0.01—that would’ve been wrong by many orders of magnitude as a prior!) (This is essentially @clone of saturn’s explanation, so see his sibling comment for more on this point.)

Yeah, I think I described P(A|B) when trying to describe the sensitivity, you are right that whether aliens actually abduct people given Bob experienced aliens abducting him is P(A|B). It’s possible I need to retract the whole section and example.

I agree. But I don’t think that you should discard the text entirely, because it seems to me that there is actually a lesson here.

I have had this experience many times: someone (sometimes on this very website) will say something like, “I know for a fact that X; my experience proves it to me beyond any doubt; I accept that my account of it won’t convince you of X, but I at least am certain of it”.

And what I often think in such cases (but perhaps too rarely say) is:

“But you shouldn’t be certain of it. It’s not just that I don’t believe X, merely based on your experience. It’s that you shouldn’t believe X, merely based on your experience. You, yourself, have not seen nearly enough evidence to convince you of X—if you were being a proper Bayesian about it. Not just my, but your conclusion, should be that, actually, X is probably false. Your experience is insufficient to convince me, but it should not have convinced you, either!”

(EDIT: For example.)

(This is related to something that Robyn Dawes talks about in Rational Choice in an Uncertain World, when he says that people are often too eager to learn from experience.)

This is also related to what E. T. Jaynes calls “resurrection of dead hypotheses”. If you have an alien abduction experience, then this should indeed raise your probability estimate of aliens existing and abducting people. But it should also raise your probability estimate of you being crazy and having hallucinations (to take one example). And since the latter was much more probable than the former to begin with, and the evidence was compatible with both possibilities, observing the evidence cannot result in our coming to believe the former rather than the latter. As Jaynes says (in reference to his example of whether evidence of psychic powers should make one believe in psychic powers):

…Indeed, the very evidence which the ESPers throw at us to convince us, has the opposite effect on our state of belief; issuing reports of sensational data defeats its own purpose. For if the prior probability of deception is greater than that of ESP, then the more improbable the alleged data are on the null hypothesis of no deception and no ESP, the more strongly we are led to believe, not in ESP, but in deception. For this reason, the advocates of ESP (or any other marvel) will never succeed in persuading scientists that their phenomenon is real, until they learn how to eliminate the possibility of deception in the mind of the reader.

The mammogram problem is different because you're only trying to determine whether a specific woman has cancer, not whether cancer exists at all as a phenomenon. If Bob was abducted by aliens, it implies that alien abduction is real, but the converse isn't true. You either need to do two separate Bayesian updates (what's the probability that Bob was abducted given his experience, and then what's the probability of aliens given the new probability that Bob was abducted), or you need a joint distribution covering all possibilities (Bob not abducted, aliens not real; Bob not abducted, aliens real; Bob abducted, aliens real).

Hrm. Maybe the slip is accidentally switching whether I'm looking for "do aliens abduct people, given Bob experienced being abducted" vs "was Bob's abduction real, given Bob experienced being abducted."

But if Bob's abduction was real, then aliens do abduct people. It would still count even if his was the only actual abduction in the history of the human race. Seems like this isn't the source of the math not working?

Semi-transitive trust is also how scientists update their beliefs, and therefore how we form scientific conclusions.

So the dilemma about how to update in your examples also govern our understanding of cutting edge scientific issues. I'm pretty sure of that after spending a couple of decades as a research scientist. I suspect this is also true of our understandings of politics, business, and everything else that makes the modern world turn. Including our understanding of the interlocking questiosn underlying the alignment problem.

In the case of scientific inquiry, the other scientists have tried to state more of why they've formed their beliefs; they cite statistics, methods, and other studies. But you can't understand all of their methods; they're never literally all stated, and you know you're misunderstanding even some of the ones that are stated. And you can't possibly read and fully understand all of the citations they give, unless this is exactly your research are, and you're addressing exactly the same question. Even then, you probably don't have time to read every cited study in detail, let alone ask the authors in person to clarify.

Add to these problems the blurry line between lying and biased reporting.

I think understanding this, as well as its place in our epistemics (as addressed in the excellent Truthseeking is the ground in which other principles grow is critical for making progress on complex questionsn.

Another error: you have P(ban) = (0.3 · 0.005) + (0.001 · 0.995) but it should be P(ban) = (0.3 · 0.005) + (0.01 · 0.995) (0.001 would be 0.1% false positive rate, not 1% as you stipulate). This results in P(problem|ban) = 14.85% rather than 13.1%.

(Also, I am actually not sure how you got 13.1%. With the aforementioned error, the result would be 60.1%…)

You are correct I added an extra 0, writing (0.3 · 0.005) + (0.001 · 0.995) when I meant (0.3 · 0.005) + (0.01 · 0.995). That's a transcription error, thank you for catching it.

I'm not sure how you're getting 14.85% or 60.1% though? I just checked, and I think those numbers do wind up at ~13.1%, not 14.85%.

Yep, my mistake. I’m not sure either! Math is hard, it seems…

EDIT: That is, I’m not sure how I got 14.85%—that was, like, I pressed the wrong calculator keys, or something. But 60.1% is like this:

(0.3 · 0.005) / ((0.3 · 0.005) + (0.01 · 0.995)) = ~0.60120 = 60.12%

Adversarial action makes this at least an order of magnitude worse.  If Carla has to include the chance that Bob could be lying (for attention, for humor, or just pathologically) about his experiences or his history of drug use or hallucinations, she makes an even smaller update.  This is especially difficult in group-membership or banning discussions, because LOTS of people lie (or just focus incorrectly on different evidence) for status and for irrelevant-beef reasons.

I don't think there is a solution, other than to acknowledge that such decisions will always be a balance of false-positive and false-negative, and it's REQUIRED to ban some innocent (-ish) people in order to protect against the likely-but-not-provably-harmful.

I agree adversarial action makes this much worse.

I think Bob and Carla's problem isn't really whether Bob is lying or not. If they knew for an absolute fact Bob wasn't speaking things he knew to be factually untrue, Carla still has to sort through misunderstanding (maybe Bob's talking about a LARP?) and drug use (maybe Bob forgot whether he took LSD the way I forget whether I've had coffee sometimes?) and psychotic breaks. I wouldn't usually count any of those as "lying" in the relevant sense; Bob's wrong, but he's accurately reporting his experiences as best he can.

I don't have a solution for the group membership case, which I think of as a special case of the reputation problem. I'm trying to point out a couple failure modes; one where you don't realize a bunch of your information actually has a single source and should be counted once, and one where you don't actually get or incorporate reputation information at all.

This problem doesn't seem to be about trust at all, it seems to be about incomplete sharing of information. It seems weird to me to say Carla doesn't completely trust Bob's account if she is 100% sure he isn't lying.

The sensitivity of the test - that aliens actually abduct people, given someone is telling her aliens abducted him - is 2.5% since she doesn't really know his drug habits and hasn't ruled out there's a LARP she's missing the context for.

I would describe this not as Carla not trusting Bob, but as her not having all of Bob's information - Bob could just tell her that he doesn't use drugs, or that he isn't referring to a LARP, or any other things he knows about himself that Carla doesn't that are causing her sensitivity to be lower, until their probabilities are the same. And, of course, if this process ends with Carla having the same probabilities as Bob, and Carla does the same with Dean, he will have the same probabilities as Bob as well.

I think this satisfies Aumann's Agreement Theorem.

Well, if it does then Bob and Carla definitely have the same probabilities; that's what the problem says, after all.