There are several compromises I made for the sake of getting the idea across as simply as I could.
I think the graduate-level-textbook version of this would be much more clear about what the quotes are doing. I was tempted to not even include the quotes in the mathematical expressions, since I don’t think I’m super clear about why they’re there.
I totally ignored the difference between P(A|B) (probability conditional on B) and PB(A) (probability after learning B).
I neglect to include quantifiers in any of my definitions; the reader is left to guess which things are implicitly universally quantified.
I think I do prefer the version I wrote, which uses P(A|B) rather than PB(A), but obviously the English-language descriptions ignore this distinction and make it sound like what I really want is PB(A).
It seems like the intention is that P1 “learns” or “hears about” P2’s belief, and then P1updates (in the above Bayesian inference sense) to have a new P1,after that has the consistency condition with P2.
Obviously we can consider both possibilities and see where that goes, but I think maybe the conditional version makes more sense as a notion of whether you right now endorse something. A conditional probability is sort of like a plan for updating. You won’t necessarily follow the plan exactly when you actually update, but the conditional probability is your best estimate.
To throw some terminology out there, let’s call my thing “endorsement” and a version which uses actual updates rather than conditionals “deference” (because you’d actually defer to their opinions if you learn them).
You can know whether you endorse something, since you can know your current conditional probabilities (to within some accuracy, anyway). It is harder to know whether you defer to something, since in the case where updates don’t equal conditionals, you must not know what you are going to update to. I think it makes more sense to define the intentional stance in terms of something you can more easily know about yourself.
Using endorsement to define agency makes it about how you reason about specific hypotheticals, whereas using deference to try and define agency would make it about what actually happens in those hypotheticals (ie, how you would actually update if you learned a thing). Since you might not ever get to learn that thing, this makes endorsement more well-defined than deference.
Bayes’ theorem is the statement about P(A|B), which is true from the axioms of probability theory for any A and B whatsoever.
I actually prefer the view of Alan Hajek (among others) who holds that P(A|B) is a primitive, not defined as in Bayes’ ratio formula for conditional probability. Bayes’ ratio formula can be proven in the case where P(B)>0, but if P(B)=0 it seems better to say that conditional probabilities can exist rather than necessarily being undefined. For example, we can reason about the conditional probability that a meteor hits land given that it hits the equator, even if hitting the equator is a measure zero event. Statisticians learn to compute such things in advanced stats classes, and it seems sensible to unify such notions under the formal P(A|B) rather than insisting that they are technically some other thing.
By putting ‘‘P2(X)=p" in the conditional, you’re saying that it’s an event on Ω1, a thing with the same type as X. And it feels like that’s conceptually correct, but also kind of the hard part. It’s as if P1 is modelling P2 as an agent embedded into Ω1.
Right. This is what I was gesturing at with the quotes. There has to be some kind of translation from P2(X)=p (which is a mathematical concept ‘outside’ Ω1) to an event inside Ω1. So the quotes are doing something similar to a Goedel encoding.
While trying to understand the equations, I found it easier to visualize P1 and P2 as two separate distributions on the same Ω, where endorsement is simply a consistency condition. For belief consistency, you would just say that P1 endorses P2 on event X if P1(X)=P2(X).
But that isn’t what you wrote; instead you wrote thing this with conditioning on a quoted thing. And of course, the thing I said is symmetrical between P1 and P2, whereas your concept of endorsement is not symmetrical.
The asymmetry is quite important. If we could only endorse things that have exactly our opinions, we could never improve.
There are several compromises I made for the sake of getting the idea across as simply as I could.
I think the graduate-level-textbook version of this would be much more clear about what the quotes are doing. I was tempted to not even include the quotes in the mathematical expressions, since I don’t think I’m super clear about why they’re there.
I totally ignored the difference between P(A|B) (probability conditional on B) and PB(A) (probability after learning B).
I neglect to include quantifiers in any of my definitions; the reader is left to guess which things are implicitly universally quantified.
I think I do prefer the version I wrote, which uses P(A|B) rather than PB(A), but obviously the English-language descriptions ignore this distinction and make it sound like what I really want is PB(A).
Obviously we can consider both possibilities and see where that goes, but I think maybe the conditional version makes more sense as a notion of whether you right now endorse something. A conditional probability is sort of like a plan for updating. You won’t necessarily follow the plan exactly when you actually update, but the conditional probability is your best estimate.
To throw some terminology out there, let’s call my thing “endorsement” and a version which uses actual updates rather than conditionals “deference” (because you’d actually defer to their opinions if you learn them).
You can know whether you endorse something, since you can know your current conditional probabilities (to within some accuracy, anyway). It is harder to know whether you defer to something, since in the case where updates don’t equal conditionals, you must not know what you are going to update to. I think it makes more sense to define the intentional stance in terms of something you can more easily know about yourself.
Using endorsement to define agency makes it about how you reason about specific hypotheticals, whereas using deference to try and define agency would make it about what actually happens in those hypotheticals (ie, how you would actually update if you learned a thing). Since you might not ever get to learn that thing, this makes endorsement more well-defined than deference.
I actually prefer the view of Alan Hajek (among others) who holds that P(A|B) is a primitive, not defined as in Bayes’ ratio formula for conditional probability. Bayes’ ratio formula can be proven in the case where P(B)>0, but if P(B)=0 it seems better to say that conditional probabilities can exist rather than necessarily being undefined. For example, we can reason about the conditional probability that a meteor hits land given that it hits the equator, even if hitting the equator is a measure zero event. Statisticians learn to compute such things in advanced stats classes, and it seems sensible to unify such notions under the formal P(A|B) rather than insisting that they are technically some other thing.
Right. This is what I was gesturing at with the quotes. There has to be some kind of translation from P2(X)=p (which is a mathematical concept ‘outside’ Ω1) to an event inside Ω1. So the quotes are doing something similar to a Goedel encoding.
The asymmetry is quite important. If we could only endorse things that have exactly our opinions, we could never improve.