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David_Bolin comments on 0 And 1 Are Not Probabilities - Less Wrong
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Comments (128)
This article is largely incoherent. The main justification is the abuse of an invalid transformations: y=x/(1-x) is not the bijection that he asserts it is, because it's not a function that maps [0,1] onto R. It's a function that maps [0,1] onto [1,\intfy] as a subset of the topological closure of R. And that's okay, but you can't say "well I don't like the topological closure of R, so I'll just use R and claim that 1 is where the problem is."
Additionally, his discussion of log odds and such is perfectly fine, but ignores the fact that there are places where you do need to have an odds of 0:1, or a log odds of negative infinity. Probability theory stops working when you throw out 0 and 1, it's as simple as that.
Even if you don't want to handle tautologies or contradictions, there are other ways to get P(X)=0 or 1. The probability that a real number chosen uniformly from the real interval [0,1] is 0. It has to be. It's a provable fact under ZFC and to decide otherwise is to say that you're more attached to the idea of 0 and 1 not being probabilities than you are to the fact that mathematics is consistent and if you really believe that, well, there's absolutely nothing I have to say to you.
This is one of those situations where EY just demonstrates he knows very little mathematics.
Eliezer isn't arguing with the mathematics of probability theory. He is saying that in the subjective sense, people don't actually have absolute certainty. This would mean that mathematical probability theory is an imperfect formalization of people's subjective degrees of belief. It would not necessarily mean that it is impossible in principle to come up with a better formalization.
Errr... as I read EY's post, he is certainly talking about the mathematics of probability (or about the formal framework in which we operate on probabilities) and not about some "subjective sense".
The claim of "people don't actually have absolute certainty" looks iffy to me, anyway. The immediate two questions that come to mind are (1) How do you know? and (2) Not even a single human being?
I think he's just acknowledging the minute(?) possibility that our apparently flawless reasoning could have a blind spot. We could be in a Matrix, or have something tampering with our minds, etcetera, such that the implied assertion:
If this appears absolutely certain to me
Then it must be true
is indefensible.
There are two different things.
David_Bolin said (emphasis mine): "He is saying that in the subjective sense, people don't actually have absolute certainty." I am interpreting this as "people never subjectively feel they have absolute certainty about something" which I don't think is true.
You are saying that from an external ("objective") point of view, people can not (or should not) be absolutely sure that their beliefs/conclusions/maps are true. This I easily agree with.
It should probably be defined by calibration: do some people have a type of belief where they are always right?
Self-referential and anthropic things would probably qualify, e.g. "I believe I exist".
You can phrase statements of logical deduction such that they have no premises and only conclusions. If we let S be the set of logical principles under which our logical system operates and T be some sentence that entails Y, then S AND T implies Y is something that I have absolute certainty in, even if this world is an illusion, because the premise of the implication contains all the rules necessary to derive the result.
A less formal example of this would be the sentence: If the rules of logic as I know them hold and the axioms of mathematics are true, then it is the case that 2+2=4
The way I view that statement is: "In our formalization, agents with absolutely certain beliefs cannot change those beliefs, we want our formalization to capture our intuitive sense of how an ideal agent would update its beliefs, a formalization with a quality of fanaticism does not capture our intuitive sense of how an ideal agent would update its beliefs, therefore we do not want a quality of fanaticism."
And what state of the world would correspond to the statement "Some people have absolute certainty." ? Do you think that we can take some highly advanced and entirely fictional neuroimaging technology, look at a brain and meaningfully say, "There's a belief with probability 1." ?
And on the other hand, I'm not afraid to talk about folk certainty, where the properties of an ideal mathematical system are less relevant, where everyone can remain blissfully logically uncertain to the fact that beliefs with probability 1 and 0 imply undesirable consequences in formal systems that possess them, and say things like "I believe that absolutely." I am not afraid to say something like, "That person will not stop believing that for as long as he lives," and mean that I predict with high confidence that that person will not stop believing that for as long as he lives.
And once you believe that the formalization is trying to capture our intuitive sense of an ideal agent, and decide whether or not that quality of fanaticism captures it, and decide whether or not you're going to be a stickler about folk language, then I don't think that any question or confusion around that claim remains.
People are not "ideal agents". If you specifically construct your formalization to fit your ideas of what an ideal agent should and should not be able to do, this formalization will be a poor fit to actual, live human beings.
So either you make a system for ideal agents -- in which case you'll still run into some problems because, as has been pointed out upthread, standard probability math stops working if you disallow zeros and ones -- or you make a system which is applicable to our imperfect world with imperfect humans.
I don't see why both aren't useful. If you want a descriptive model instead of a normative one, try prospect theory.
I just don't see this article as an axiom that says probabilities of 0 and 1 aren't allowed in probability theory. I see it as a warning not to put 0s and 1s in your AI's prior. You're not changing the math so much as picking good priors.
If we're asking what the author "really meant" rather than just what would be correct, it's on record.
I... can't really recommend reading the entire thread at the link, it's kind of flame-war-y and not very illuminating.
I think the issue at hand is that 0 and 1 aren't special cases at all, but very important for the math of probability theory to work (try and construct a probability measure where some subset doesn't have probability 1 or 0).
This is incredibly necessary for the mathematical idea of probability ,and EY seems to be confusing "are 0 and 1 probabilities relevant to Bayesian agents?" with "are 0 and 1 probabilities?" (yes, they are, unavoidably, not as a special case!).
It seems that EY position boils down to
And that's a weak claim. EY's ideas of what is "mentally healthier" are, basically, his personal preferences. I, for example, don't find any mental health benefits in thinking about one over googolplex probabilities.
Cromwell's Rule is not EY's invention, and relatively uncontroversial for empirical propositions (as opposed to tautologies or the like).
If you don't accept treating probabilities as beliefs and vice versa, then this whole conversation is just a really long and unnecessarily circuitous way to say "remember that you can be wrong about stuff".
The part that is new compared to Cromwell's rule is that Yudkowsky doesn't want to give probability 1 to logical statements (53 is a prime number).
Because he doesn't want to treat 1 as a probability, you can't expect complete sets of events to have total probability 1, despite them being tautologies. Because he doesn't want probability 0, how do you handle the empty set? How do you assign probabilities to statements like "A and B" where A and B are logical exclusive? (the coin lands heads AND the coin lands tails).
Removing 0 and 1 from the math of probability breaks most of the standard manipulations. Again, it's best to just say "be careful with 0 and 1 when working with odds ratios."
Nobody is saying EY invented Cromwell's Rule, that's not the issue.
The issue is that "0 and 1 are not useful subjective certainties for a Bayesian agent" is a very different statement than "0 and 1 are not probabilities at all".
You're right, I misread your sentence about "his personal preferences" as referring to the whole claim, rather than specifically the part about what's "mentally healthy". I don't think we disagree on the object level here.
Of course if no one has absolute certainty, this very fact would be one of the things we don't have absolute certainty about. This is entirely consistent.