Maybe 1) is where I have a fundamental difference.
Given evidence A, a Bayesian update considers how well evidence A was predicted.
There is no additional update due to how well ¬A being false was predicted. Even if ¬A is split into sub-categories, it isn’t relevant as that evidence has already been taken into account when we updated based on A being true.
r.e. 2) 50:25:0:0 gives a worse expected value than 50:50:0:0 as although my score increases if A is true, it decreases by more if B is true (assuming 50:50:0:0 is my true belief)
r.e. 3) I think it’s important to note that I’m assuming that exactly 1 of A or B or C or D is the correct answer. Therefore that the probabilities should add up to 100% to maximise your expected score (otherwise it isn’t a proper scoring rule).
Maybe 1) is where I have a fundamental difference.
Given evidence A, a Bayesian update considers how well evidence A was predicted.
There is no additional update due to how well ¬A being false was predicted. Even if ¬A is split into sub-categories, it isn’t relevant as that evidence has already been taken into account when we updated based on A being true.
r.e. 2) 50:25:0:0 gives a worse expected value than 50:50:0:0 as although my score increases if A is true, it decreases by more if B is true (assuming 50:50:0:0 is my true belief)
r.e. 3) I think it’s important to note that I’m assuming that exactly 1 of A or B or C or D is the correct answer. Therefore that the probabilities should add up to 100% to maximise your expected score (otherwise it isn’t a proper scoring rule).