The Nonlinear Library LW - Intransitive Trust by Screwtape
May 27, 2024
17:00
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Intransitive Trust, published by Screwtape on May 27, 2024 on LessWrong.
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.
(I am not a math major, and if there's a consensus in the comments that I'm using the wrong term here 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.
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 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).) 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(AB)=P(BA)P(A)P(B)
P(aliensexperiences)=P(experiencesaliens)P(aliens)P(experiences)
P(Experiences)=P(ExperiencesAliens)P(Aliens)+P(ExperiencesAliens)P(Aliens)
P(Experiences)=(0.050.01)+(0.0010.99)
P(Experiences)=0.00149
P(AB)=.05.01.00149
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 ot...
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.
(I am not a math major, and if there's a consensus in the comments that I'm using the wrong term here 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.
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 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).) 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(AB)=P(BA)P(A)P(B)
P(aliensexperiences)=P(experiencesaliens)P(aliens)P(experiences)
P(Experiences)=P(ExperiencesAliens)P(Aliens)+P(ExperiencesAliens)P(Aliens)
P(Experiences)=(0.050.01)+(0.0010.99)
P(Experiences)=0.00149
P(AB)=.05.01.00149
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 ot...