(Edit: I may have been misinterpreting what you meant by “geometric mean of probabilities.” If you mean “take the geometric mean of probabilities of all events and then scale them proportionally to add to 1″ then I think that’s a pretty good method of aggregating probabilities. The point i make below is that the scaling is important.)
I think taking the geometric mean of odds makes more sense than taking the geometric mean of probabilities, because of an asymmetry arising from how the latter deals with probabilities near 0 versus probabilities near 1.
Concretely, suppose Alice forecasts an 80% chance of rain and Bob forecasts a 99% chance of rain. Those are 4:1 and 99:1 odds respectively, and if you take the geometric mean you’ll get an aggregate 95.2% chance of rain.
Equivalently, Alice and Bob are forecasting a 20% chance and a 1% chance of no rain—i.e. 1:4 and 1:99 odds. Taking the geometric mean of odds gives you a 4.8% chance of no rain—checks out.
Now suppose we instead take a geometric mean of probabilities. The geometric mean of 80% and 99% is roughly 89.0%, so aggregating Alice’s and Bob’s probabilities of rain in this way will give 89.0%.
On the other hand, aggregating Alice’s and Bob’s probabilities of no rain, i. e. taking a geometric mean of 20% and 1%, gives roughly 4.5%.
This means that there’s an inconsistency with this method of aggregation: you get an 89% chance of rain and a 4.5% chance of no rain.
(Edit: I may have been misinterpreting what you meant by “geometric mean of probabilities.” If you mean “take the geometric mean of probabilities of all events and then scale them proportionally to add to 1″ then I think that’s a pretty good method of aggregating probabilities. The point i make below is that the scaling is important.)
I think taking the geometric mean of odds makes more sense than taking the geometric mean of probabilities, because of an asymmetry arising from how the latter deals with probabilities near 0 versus probabilities near 1.
Concretely, suppose Alice forecasts an 80% chance of rain and Bob forecasts a 99% chance of rain. Those are 4:1 and 99:1 odds respectively, and if you take the geometric mean you’ll get an aggregate 95.2% chance of rain.
Equivalently, Alice and Bob are forecasting a 20% chance and a 1% chance of no rain—i.e. 1:4 and 1:99 odds. Taking the geometric mean of odds gives you a 4.8% chance of no rain—checks out.
Now suppose we instead take a geometric mean of probabilities. The geometric mean of 80% and 99% is roughly 89.0%, so aggregating Alice’s and Bob’s probabilities of rain in this way will give 89.0%.
On the other hand, aggregating Alice’s and Bob’s probabilities of no rain, i. e. taking a geometric mean of 20% and 1%, gives roughly 4.5%.
This means that there’s an inconsistency with this method of aggregation: you get an 89% chance of rain and a 4.5% chance of no rain.
Ohhhhhhhhhhhhhhhhhhhhhhhh
I had not realized, and this makes so much sense.