It’s true that the prior should not be “A and B are independent”. But shouldn’t symmetries of how they may be dependent give essentially the same result as assuming independence? Similar as to how any symmetric prior for how a coin is biased gives the same results for a prediction of probability of heads -- 1⁄2.
I don’t think independence is a good way to analyze things when the probabilities are near zero or one. Independence is just P[A] P[B] = P[AB]. If P[A] or P[B] are near zero or one, this is automatically “nearly true”.
Put another way, two observation of (A, B) give essentially no information about dependence by themselves. This is encoded into ratios between the four possibilities.
It’s true that the prior should not be “A and B are independent”. But shouldn’t symmetries of how they may be dependent give essentially the same result as assuming independence? Similar as to how any symmetric prior for how a coin is biased gives the same results for a prediction of probability of heads -- 1⁄2.
I don’t think independence is a good way to analyze things when the probabilities are near zero or one. Independence is just P[A] P[B] = P[AB]. If P[A] or P[B] are near zero or one, this is automatically “nearly true”.
Put another way, two observation of (A, B) give essentially no information about dependence by themselves. This is encoded into ratios between the four possibilities.