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.
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.