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taw comments on 5 Axioms of Decision Making - Less Wrong
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If you answered 1 to the first, and anything but 0 or 1 to the second, you're inconsistent. If you're unwilling to answer the second, you just broke your axioms.
Subjective probability allows logical uncertainty.
Subjective probabilities are inconsistent in any model which includes Peano arithmetic by straightforward application of Gödel's incompleteness theorems, which is essentially any non-finite model.
Most people here seem to be extremely unwilling to admit that probabilities and uncertainty are not the same thing.
Could you explain why this is true, please?
Let X() be a consistent probability assignment (function from statement to probability number).
Let Y() be a probability assignment including: Y(2+2=5) = X(Y is consistent), and otherwise Y(z)=X(z)
What's X(Y is consistent)?
If X(Y is consistent)=1, then Y(2+2=5)=1, and Y is blatantly inconsistent, and so is X is inconsistent according to basic laws of mathematics.
If X(Y is consistent)=0, then Y(2+2=5)=0=X(2+2=5), and by definition X=Y, so X is inconsistent according to itself.
What if X(Y is consistent)=0.5? Then Y(2+2=5) = 0.5, and Y might or might not be inconsistent.
Another solution is of course to let X be incomplete, and refuse to assign X(Y is consistent). In fact, that would be the sensible thing to do. X can never be a function from ''all'' statements to probabilities, it's domain should only include statements strictly smaller than X itself.
If Y(2 + 2 = 5) = 0.5, Y is still blatantly inconsistent, so that won't help.
I think your second point might be right, though. Isn't it the case that the language of first-order arithmetic is not powerful enough to refer to arbitrary probability assignments over its statements? After all, there are an uncountable number of such assignments, and only a countable number of well-formed formulas in the language. So I don't see why a probability assignment X in a model that includes Peano arithmetic must also assign probabilities to statements like "Y is consistent".
If you let X be incomplete like twanvl suggests, then you pretty much agree with my position of using probability as a useful tool, and disagree with Bayesian fundamentalism.
Getting into finer points of what is constructible or provable in what language is really not a kind of discussion we could usefully have within confines of lesswrong comment boxes, since we would need to start by formalizing everything far more than we normally do. And it wouldn't really work, it is simply not possible to escape Goedel's incompleteness theorem if you have something even slightly powerful and non-finite, it will get you one way or another.
I'm possibly being obtuse here, but I still don't see the connection to the incompleteness theorem. I don't deny that any consistent theory capable of expressing arithmetic must be incomplete, but what does that have to do with the argument you offered above? That argument doesn't hinge on incompleteness, as far as I can see.
How is a distribution useful if it refuses to answer certain questions? I think I'm misunderstanding something you said, since I think that the essence of Bayesianism is the idea that probabilities must be used to make decisions, while you seem to be contrasting these two things.
This is slightly exaggerated. The theory of real numbers is non-finite and quite powerful, but it has a complete axiomatization.
What does it mean for this function to be "consistent"? What kinds of statements do you allow?
If "probability assignment" is a mapping from statements (or Goedel numbers) to the real interval [0,1], it's not a given that Y, being a "probability assignment", is definable, so that you can refer to it in the statement "Y is consistent" above.
That's not Goedelian at all, it's a variant of Russell's paradox and can be excluded by an analogue of the theory of types (which would make Y an illegally self-referential probability assignment).
I can't speak for anyone else, but for my part that's because I rarely if ever see the terms used consistently to describe different things. That may not be true of mathematicians, but very little of my language use is determined by mathematicians.
For example, given questions like:
1) When I say that the coin I'm about to flip has an equal chance of coming up heads or tails, am I making a statement about probability or uncertainty?
2) When I say that the coin I have just flipped, but haven't yet looked at, has an equal chance of having come up heads or tails, am I making a statement about probability or uncertainty?
3) When I say that the coin I have just looked at has a much higher chance of having come up heads rather than tails, but you haven't looked at the coin yet and you say at the same time that it has an equal chance of having come up heads or tails, are we both making a statement about the same thing, and if so which thing is it?
...I don't expect consistent answers from 100 people in my linguistic environment. Rather I expect some people will answer "uncertainty" in all three cases, other people will answer "probability", still others will give neither answer. Some might even say that I'm talking about "probability" in case 1, "uncertainty" in case 2, and that in case 3 I'm talking about uncertainty and you're talking about probability.
In that kind of linguistic environment, it's safest to treat the words as synonyms. If someone wants to talk to me about the difference between two kinds of systems in the world, the terms "probability" and "uncertainty" aren't going to be very useful for doing so unless they first provide two definitions.
People tell me otherwise.
I don't know if that actually solves the problem. Nor do I know if it makes sense to claim that understanding the two meanings of a Gödel statement, and the link between them, puts you in a different formal system which can therefore 'prove' the statement without contradiction. But it seems to me this accounts for what we humans actually do when we endorse the consistency of arithmetic and the linked mathematical statements. We don't actually have the brains to write a full Gödel statement for our own brains and thereby produce a contradiction.
In your example below, X(Y is consistent) might in fact be 0.5 because understanding what both systems say might put us in Z. Again, this may or may not solve the underlying problem. But it shouldn't destroy Bayesianism to admit that we learn from experience.
Did you mean to say incomplete (eg, implying that some small class of bizarrely constructed theorems about subjective probability can't be proven or disproven)?
Because the standard difficulties that Godel's theorem introduces to Peano arithmetic wouldn't render subjective probabilities inconsistent (eg, no theorems about subjective probability could be proven).
If (basic laws = axioms and inference rules), meaning of "true" needs clarification.
0.64 (Here, by "true" I mean "can be proven in Peano arithmetics".)
Then you're enitrely inconsistent, since P(Collatz sequence for k converges) is either 0 or 1 for all k by basic laws of mathematics, and P(Collatz conjecture is true) equals product of these, and by basic laws of mathematics can only be 0 or 1.
Why had you chosen Collatz conjecture to illustrate the fact (which already has been discussed several times) that uncertainty about mathematical statements introduces inconsistency of some sort? I am equally willing to put p = 0.1 to the statement "last decimal digit of 1543! is 7", although in fact this is quite easy to check. Just I don't want to spend time checking.
If for consistency you demand that subjective probabilities assigned to logically equivalent propositions must be equal (I don't dispute that it is sensible to include that to definition of "consistent"), then real people are going to be inconsistent, since they don't have enough processing power to check for consistency. This is sort of trivial. People hold inconsistent beliefs all the time, even when they don't quantify them by probabilities.
If you point to some fine mathematical problems with "ideal Bayesian agents", then I don't see how it is relevant in context of the original post.
Edit: by the way,
sounds frequentistish.
What probabilities are are you willing to assign to statements:
Bayesian probabilities don't give you any anchoring to reality, they only give you consistency.
If you're willing to abandon consistency as well, they give you precisely nothing whatsoever.
Probabilities are a tool for talking about uncertainty, they are not uncertainty, to think otherwise is a ridiculous map-territory confusion.
As ad hominem attacks go, that's an interesting one.
If there's one possible universe where Collatz conjecture is true/false, it is true/false is all other possible universes as well. There are no frequencies there, it's just pure fact of logic.
Updated. (Didn't occur to me it would be so easy.)
It is unnecessarily black-and-white point of view on consistency. I can improve my consistency a lot without becoming completely consistent. In practice we all compartmentalise.
I did certainly not dispute that (if I understand correctly what you mean, which I am not much sure about).
The point was, subjective probability is a degree of belief in the proposition; saying "it must be either 0 or 1 by laws of mathematics" rather implies that it is an objective property of the proposition. This seems to signal that you use a non-subjectivist (not necessarily frequentist, my fault) interpretation of probability. We may be then talking about different things. Sorry for ad hominem impression.
1 to the first for reasonable definitions of "true." .8 to the second- it seems like the sort of thing that should be true.
To assess the charge of inconsistency, though, we have unpack what you mean by that. Do you mean that I can't see the mathematical truth of a statement without reasoning through it? Then, yes, I very much agree with you. That is not a power I have. (My reasoning is also finite; I doubt I will solve the Collatz conjecture.)
But what I mean by an uncertainty of .8 is not "in the exterior world, a die is rolled such that the Collatz conjecture is true in 80% of universes but not the rest." Like you point out, that would be ridiculous. I'm not measuring math; I'm measuring my brain. What I mean is "I would be willing to wager at 4-1 odds that the Collatz conjecture is true for sufficiently small dollar amounts." Inconsistency, to me, is allowing myself to be Dutch Booked- which those two probabilities do not do.
You can be "Dutch booked" by someone who can solve the conjecture. (I am not sure whether this can be referred to as Dutch booking, but it would be the case where you both would have access to the same information and one would be in a better position due to imperfections in the other's reasoning.)
It seems also a bit like the sort of thing that might be undecidable.
I'm pretty sure that a Dutch Book is only a Dutch Book if it's pure arbitrage- that is, you beat someone using only the odds they publish. If you know more than someone else and win a bet against them, that seems different.
Quite possibly. I'm not a good judge of mathematical truth- I tend to be more trusting than I should be. It looks to me like if you can prove "every prime can be expressed as the output of algorithm X", where X is some version of the Collatz conjecture in reverse, then you're done. (Heck, that might even map onto the Sieve of Eratosthenes.) That it isn't solved already drops my credence down from ~.95 to ~.8.
Yes. Crudely speaking they have to be stupid, not just ignorant!
Not being able to decide upon the Collatz conjecture is stupidity, not ignorance. A very widespread sort of stupidy, but still.
Grandparent is self contained and entirely Collatz-independent.
They publish probability of axioms of arithmetics being roughly 1 and probability of Collatz conjecture being 0.8, you see that the conjecture is logically equivalent to the axioms and thus that their odds are mutually inconsistent. You don't "know" more in the sense of having observed more evidence. (I'd agree that this is a tortured interpretation of Dutch booking, but it's probably what you get if you systematically distinguish external evidence from own reasoning.)
While this sort of thing is interesting, I really don't see its relevance to practical decision making methods as discussed in this post. In fact, the OP even has an escape clause ‘important to your decision’ that applies perfectly here. (The Collatz Conjecture is not important to your decision, almost always in the real world.)