An explicit case where I think it’s important to arithmean over your subjective distribution of beliefs:
coin A is fair
coin B is either 2% heads or 98% heads, you don’t know
you lose if either comes up tails.
So your p(win) is “either 1% or 49%”.
I claim the FF should push the button that pays us $80 if win, -$20 if lose, and in general make action decisions consistent with a point estimate of 25%. (I’m ignoring here the opportunity to seek value of information, which could be significant!).
It’s important not to use geomean-of-odds to produce your actions in this scenario; that gives you ~9.85%, and would imply you should avoid the +$80;-$20 button, which I claim is the wrong choice.
An explicit case where I think it’s important to arithmean over your subjective distribution of beliefs:
coin A is fair
coin B is either 2% heads or 98% heads, you don’t know
you lose if either comes up tails.
So your p(win) is “either 1% or 49%”.
I claim the FF should push the button that pays us $80 if win, -$20 if lose, and in general make action decisions consistent with a point estimate of 25%. (I’m ignoring here the opportunity to seek value of information, which could be significant!).
It’s important not to use geomean-of-odds to produce your actions in this scenario; that gives you ~9.85%, and would imply you should avoid the +$80;-$20 button, which I claim is the wrong choice.