I’ve updated the text to replace ‘geometric mean’ with ‘geometric mean of odds’ everywhere it occurs. Thanks so much for the close reading and spotting the error.
Thanks! Though it’s not so much an error as just moderately confusing communication.
As you probably already know, I think one advantage of geometric mean of odds over probabilities is that it directly addresses one of Ross’s objections:
> Consider an experiment where you flip a fair coin A. If A is heads you flip a 99%heads coin B; if A is tails you flip a 1%heads coin B. We’re interested in forming a subjective probability that B is heads.
The answer I find intuitive for p(B=heads) is 50%, which is achieved by taking the arithmetic average over worlds. The geometric average over worlds gives 9.9% instead, which doesn’t seem like “fair betting odds” for B being heads under any natural interpretation of those words. What’s worse, the geometric-mean methodology suggests a 9.9% subjective probability of tails, and then p(H)+p(T) does not add to 1.
Geomean of odds of 99% heads and 1% heads is
sqrt(99∗1):sqrt(1∗99)=1:1=50
More generally, geomean of X:Y and Y:X is 50%, and geomean of odds is equally sensitive to outlier probabilities in both directions (whereas geomean of probabilities is only sensitive to outlierly low probabilities).
Thanks! Though it’s not so much an error as just moderately confusing communication.
As you probably already know, I think one advantage of geometric mean of odds over probabilities is that it directly addresses one of Ross’s objections:
Geomean of odds of 99% heads and 1% heads is
sqrt(99∗1):sqrt(1∗99)=1:1=50
More generally, geomean of X:Y and Y:X is 50%, and geomean of odds is equally sensitive to outlier probabilities in both directions (whereas geomean of probabilities is only sensitive to outlierly low probabilities).
I agree that geomean-of-odds performs better than geomean-of-probs!
I still think it has issues for converting your beliefs to actions, but I collected that discussion under a cousin comment here: https://forum.effectivealtruism.org/posts/Z7r83zrSXcis6ymKo/dissolving-ai-risk-parameter-uncertainty-in-ai-future?commentId=9LxG3WDa4QkLhT36r