The whole debate seems to be poorly founded. There are many ways to combine probability estimates that have different properties and are suited to different situations. There is no one way that works best for every purpose and context.
In some contexts, the best estimate won’t even be within the range of the individual estimates. For example, suppose there is some binary question with prior 1:1 odds. Three other people have independent evidence regarding the question and give odds of 1:2, 1:3, and 1:4 based on their own evidence. What is your best estimate for the odds? (Hint: it’s not anywhere between 1:2 and 1:4)
I agree, and in fact I already gave almost the same example in the original post. My claim was not that averaging probabilities is always appropriate, just that it is often reasonable, and average log odds never is.
The whole debate seems to be poorly founded. There are many ways to combine probability estimates that have different properties and are suited to different situations. There is no one way that works best for every purpose and context.
In some contexts, the best estimate won’t even be within the range of the individual estimates. For example, suppose there is some binary question with prior 1:1 odds. Three other people have independent evidence regarding the question and give odds of 1:2, 1:3, and 1:4 based on their own evidence. What is your best estimate for the odds? (Hint: it’s not anywhere between 1:2 and 1:4)
I agree, and in fact I already gave almost the same example in the original post. My claim was not that averaging probabilities is always appropriate, just that it is often reasonable, and average log odds never is.