If you’re talking about math, Bayes’ theorem is true and that’s the end of that. If you’re talking about degrees of belief that real people hold—especially if you want to convince your opponent to update in a specific direction because Bayes’ theorem says so—I’d advise to use another strategy. Going meta like “you must be persuaded by these arguments because blah blah blah” gives you less bang per buck than upgrading the arguments.
What kind of math do you know in where things can be “true, and that’s the end of that”? In math, things should be provable from a known set of axioms, not chosen to be true because they feel right. Change the axioms, and you get different result.
Intuition is a good guide for finding a proof, and in picking axioms, but not much more than that. And intuitively true axioms can easily result in inconsistent systems.
The questions, “what axioms do I need to accept to prove Bayes’ Theorem?”, “Why should I believe these axioms reflect the physical universe”? and “What proof techniques do I need to prove the theorem?” are very relevant to deciding whether to accept Bayes’ Theorem as a good model of the universe.
Thank you all. It seems I perhaps haven’t phrased my question the way I thought of it.
I don’t doubt the validity of the proofs underlying Bayes’ theorem, just as I don’t doubt the validity of Euclidian geometry. The question is rather if BT/probability theory hinges on assumptions that may turn out not to be necessarily true for all possible worlds, geometries, curvatures, whatever. This turned out to be the case for Euclidian geometry, as it did for Zeno. They assumed features of the world which turned out not to be the case.
It may be that my question doesn’t even make sense, but what I was trying to convey was what apriori assumptions does BT rely on which may turn out to be dodgy in the real world?
I’m not as such trying to convince people, rather trying to understand my own side’s arguments.
Bayes’ Theorem assumes that it is meaningful to talk about subjective degrees of belief, but beyond that all you really need is basic arithmetic. I can’t imagine a universe in which subjective degrees of belief aren’t something that can be reasoned about, but that may be my failure and not reality’s.
If you’re talking about math, Bayes’ theorem is true and that’s the end of that. If you’re talking about degrees of belief that real people hold—especially if you want to convince your opponent to update in a specific direction because Bayes’ theorem says so—I’d advise to use another strategy. Going meta like “you must be persuaded by these arguments because blah blah blah” gives you less bang per buck than upgrading the arguments.
What kind of math do you know in where things can be “true, and that’s the end of that”? In math, things should be provable from a known set of axioms, not chosen to be true because they feel right. Change the axioms, and you get different result.
Intuition is a good guide for finding a proof, and in picking axioms, but not much more than that. And intuitively true axioms can easily result in inconsistent systems.
The questions, “what axioms do I need to accept to prove Bayes’ Theorem?”, “Why should I believe these axioms reflect the physical universe”? and “What proof techniques do I need to prove the theorem?” are very relevant to deciding whether to accept Bayes’ Theorem as a good model of the universe.
Bayes’ theorem doesn’t require much more than multiplication and division. Here’s some probability definitions:
P(A) = the probability of A happening P(A|B) = the probability of A happening given B has happened P(AB) = the probability of both A and B happening
For example, if A is a fair, six-sided die rolling a 4 and B is said die rolling an even, then P(A) = 1⁄6, P(A|B) = 1⁄3, P(AB) = 1⁄6.
By definition, P(A|B)=P(AB)/P(B). In words, the probability of A given B is equal to the probability of both A and B divided by the probability of B.
Solving for P(AB) tells us that:
P(B)P(A|B) = P(AB) = P(A)P(B|A)
Taking out the middle and solving for P(B) allows us to flip-flop from one-side of the given to the other.
P(A|B)=P(A)*P(B|A)/P(B)
Voila! Bayes’ Theorem is logically necessary.
I’d love to hear more reasons, but here’s one:
The fact that we find it intuitive is (via evolution) evidence that it in fact is true in this universe.
Right?
Unfortunately, there are enough exceptions to that rule that it probably only counts as weak evidence.
Thank you all. It seems I perhaps haven’t phrased my question the way I thought of it.
I don’t doubt the validity of the proofs underlying Bayes’ theorem, just as I don’t doubt the validity of Euclidian geometry. The question is rather if BT/probability theory hinges on assumptions that may turn out not to be necessarily true for all possible worlds, geometries, curvatures, whatever. This turned out to be the case for Euclidian geometry, as it did for Zeno. They assumed features of the world which turned out not to be the case.
It may be that my question doesn’t even make sense, but what I was trying to convey was what apriori assumptions does BT rely on which may turn out to be dodgy in the real world?
I’m not as such trying to convince people, rather trying to understand my own side’s arguments.
I think Kevin Van Horn’s introduction to Cox’s theorem (warning: pdf) is exactly what you’re looking for.
(If you read the article, please give me feedback on the correctness of my guess that it addresses your concern.)
Bayes’ Theorem assumes that it is meaningful to talk about subjective degrees of belief, but beyond that all you really need is basic arithmetic. I can’t imagine a universe in which subjective degrees of belief aren’t something that can be reasoned about, but that may be my failure and not reality’s.