4 ms·
P(A|B) * P(B) = P(A & B) = P(B|A) * P(A) (Under the assumption that P(B) > 0 and P(A) > 0 ): P(A|B) > P(A) iff P(A|B) * P(B) > P(A) * P(B) and so iff P(A & B)
by drdeca 3y ago
P(A|B) * P(B) = P(A & B) = P(B|A) * P(A)
(Under the assumption that P(B) > 0 and P(A) > 0 ):
P(A|B) > P(A) iff P(A|B) * P(B) > P(A) * P(B)
and so iff P(A & B) > P(A) * P(B)
and therefore, by the same reasoning in the other direction,
P(A|B) > P(A) iff P(B|A) > P(B)
(Let A and B be “the CCP is evil” and “the CCP committed [some particular wrongdoing]”)
- pphysch 3y agoP(A)=1 is an axiom of mainstream Western thought.
- drdeca 3y agoWhile, uh, no, regardless, that has no bearing on my point, which is that your point claiming that [it is reasonable to treat A as evidence for B, but irrational/religious to treat B as evidence for A], is entirely wrong[1]. [1](... with the possible exception of observing something which one previously had assigned a probability of zero, but, this ideally should only ever happen for things where one at least had a positive probability density, in which case a similar argument should still work. I didn't feel like working out all the details in the case of continuous probability distributions, as it is finicky and in any case is irrelevant to the discrete events you mentioned.)