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Perplexed comments on Dutch Books and Decision Theory: An Introduction to a Long Conversation - Less Wrong
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There is no hope for LessWrong as long as people keep conflating Perfect Bayesian and Subjective Bayesian.
Let's take Subjective Bayesian first. The problem is - Subjective Bayesian breaks basic laws of probability as a routine matter.
Take the simplest possible law of probability P(X) >= P(X and Y).
Now let's X be any mathematical theorem which you're not sure about. 1 > P(X) > 0.
Let Y be some kind of "the following proof of X is correct".
Verifying proofs is usually very simple, so very once you're asked about P(X and Y), you can confidently reply P(X and Y) = 1. Y is not a new information about the world. It is usually conjunction of trivial statements which you already assigned probability 1.
That is - there's infinite number of statements for which Subjective Bayesian will reply P(X) < P(X and Y).
For Subjective Bayesian X doesn't even have to involve any infinities, just ask a simple question about cryptography which is pretty much guaranteed to be unsolvable before heat death of the universe, and you're done.
At this point people far too often try to switch Perfect Bayesian for Subjective Bayesian.
And this is true, Perfect Bayesian wouldn't make this particular mistake, and all probabilities of mathematical theorems he'd give would be 0 or 1, no exceptions. The problem is - Perfect Bayesians are not possible due to uncomputability.
If your version of Perfect Bayesian is computable, straightforward application of Rice Theorem shows he won't be able to answer every question consistently.
If you claim some super-computable oracle version Perfect Bayesian - well, first that's already metaphysics not mathematics, but in the end, this way of working around uncomputability does not work.
At any mention of uncomputability people far too often try to switch Subjective Bayesian for Perfect Bayesian (see Eliezer's comment).
Ok, I understood that, but I still don't see what it has to do with Dutch books.
P(X) < P(X and Y) gives you a dutch book.
OH I SEE. Revelation.
You can get pwn'd if the person offering you the bets knows more than you. The only defense is to, when you're uncertain, have bets such that you will not take X or you will not take -X (EDIT: Because you update on the information that they're offering you a bet. I forgot that. Thanks JG.). This can be conceptualized as saying "I don't know"
So yeah. If you suspect that someone may know more math than you, don't take their bets about math.
Now, it's possible to have someone pre-commit to not knowing stuff about the world. But they can't pre-commit to not knowing stuff about math, or they can't as easily.
Another defense is updating on the information that this person who knows more than you is offering this bet before you decide if you will accept it.
That's not the Dutch book I was talking about.
Let's say you assign "X" probability of 50%, so you gladly take 60% bet against "X".
But you assign "X and Y" probability 90%, so you as gladly take 80% bet for "X and Y".
You just paid $1.20 for combinations of bets that can give you returns of at most $1 (or $0 if X turns out to be true but Y turns out to be false).
This is exactly a Dutch Book.
Given that they are presented at the same time (such as X is a conjecture, Y is a proof of the conjecture), yes, accepting these bets is being Dutch Booked. But upon seeing "X and Y" you would update "X" to something like 95%.
Given that they are presented in order (What bet do you take against X? Now that's locked in, here is a proof Y. What bet do you take for "X and Y"?) this is a malady that all reasoners without complete information suffer from. I am not sure if that counts as a Dutch Book.
It is trivial to reformulate this problem to X and X' being logically equivalent, but not immediately noticeable as such, and a person being asked about X' and (X and Y) or something like that.
Yes, but that sounds like "If you don't take the time to check your logical equivalencies, you will take Dutch Books". This is that same malady: it's called being wrong. That is not a case of Bayesianism being open to Dutch Books: it is a case of wrong people being open to Dutch Books.
You're very wrong here.
By Goedel's Incompleteness Theorem, there is no way to "take the time to check your logical equivalencies". There are always things that are logically equivalent that your particular method of proving, no matter how sophisticated, will not find, in any amount of time.
This is somewhat specific to Bayesianism, as Bayesianism insists that you always give a definite numerical answer.
Not being able to refuse answering (by Bayesianism) + no guarantee of self-consistency (by Incompleteness) => Dutch booking
I admit defeat. When I am presented with enough unrefusable bets that incompleteness prevents me from realising are actually Dutch Books such that my utility falls consistently below some other method, I will swap to that method.