People, Bayes-structure doesn’t require Bayes-math! Thinking about the math allows us, among other things, to pick computationally efficient approximations that are closer to normative reasoning, as exemplified by past posts like (picks at random) this one. I don’t need to have any idea what numerical probabilities I should be assigning to know that P(A&B)<=P(A), P(A|B)>=P(A), if seeing A increases P(B) then seeing ~A must decrease it, and so on. It’s a little like how (would a physicist please stop me if this is inaccurate?) a knowledge of quantum mechanics allows you to create semiclassical approximations that make better predictions than classical models but are more tractable than the genuine QM math.
As a social ideal, Science doesn’t judge you as a bad person for coming up with heretical hypotheses
Surely this is a good thing, else fear of being thought stupid would overly discourage people from raising novel hypotheses. The problem is encouraging a private, epistemic standard as lax as the social one.
It is not true in general that P(A|B) >= P(A). For example, suppose you are rolling a die that could be 6-sided (cube) or 4-sided (tetrahedron); then let A = “the die comes up showing a 6” and B = “the die is 4-sided”. We have P(A|B)=0, yet P(A) > 0 as long as P(B) < 1.
People, Bayes-structure doesn’t require Bayes-math! Thinking about the math allows us, among other things, to pick computationally efficient approximations that are closer to normative reasoning, as exemplified by past posts like (picks at random) this one. I don’t need to have any idea what numerical probabilities I should be assigning to know that P(A&B)<=P(A), P(A|B)>=P(A), if seeing A increases P(B) then seeing ~A must decrease it, and so on. It’s a little like how (would a physicist please stop me if this is inaccurate?) a knowledge of quantum mechanics allows you to create semiclassical approximations that make better predictions than classical models but are more tractable than the genuine QM math.
Surely this is a good thing, else fear of being thought stupid would overly discourage people from raising novel hypotheses. The problem is encouraging a private, epistemic standard as lax as the social one.
It is not true in general that P(A|B) >= P(A). For example, suppose you are rolling a die that could be 6-sided (cube) or 4-sided (tetrahedron); then let A = “the die comes up showing a 6” and B = “the die is 4-sided”. We have P(A|B)=0, yet P(A) > 0 as long as P(B) < 1.
He probably didn’t use standard notation here. I would read P(A|B) as P(A OR B) in this context.