In short: I still recommend using the geometric mean of odds as the default aggregation method, but I give my best guess on exceptions to this rule.
Since writing about how the geometric mean of odds compares to other forecasting aggregation methods I have received many comments asking for a more nuanced approach to choosing how to aggregate forecasts. I do not yet have a full answer to this question, but here I am going to outline my current best guess to help people with their research and give a chance to commenters to prove me wrong.
In short, here is my current best guess in the form of a flowchart:
Some explanations are in order:
- I currently believe that the geometric mean of odds should be the default option for aggregating forecasts. In the two large scale empirical evaluations I am aware of  , it surpasses the mean of probabilities and the median (*). It is also the only method that makes the group aggregate behave as a Bayesian, and (in my opinion) it behaves well with extreme predictions.
- If you are not aggregating all-considered views of experts, but rather aggregating models with mutually exclusive assumptions, use the mean of probabilities. For example, this will come up if you first compute your timelines for Transformative AI assuming it will be derived from transformer-like methods, and then assuming it will come from emulated beings, etc. In this case, .
- When the data includes poorly calibrated outliers, if it's possible exclude them and take the geometric mean. If not, we should use a pooling method resistant to outliers. The median is one such popular aggregation method.
- If there is a known bias in the community of predictors you are polling for predicting positive resolution of binary questions, you can consider correcting for this. One correction that worked on metaculus data is taking the geometric mean of the probabilities (this pulls the aggregate towards zero compared to the geometric mean of odds). Better corrections are likely to exist. [EDIT: I rerun the test recently and the geo mean of probabilities does no longer outcompete the geo mean of odds. Consequently, I no longer endorse using the geometric mean of probabilities]
- If there is a track record of underconfidence in past aggregate predictions from the community, consider extremizing the final outcome. This has been common practice in academia for a while. For example, (Satopää et al, 2014) have found good performance using extremized logodds. To choose an extremizing factor I suggest experimenting with what extremizing factors would have given you good performance in past predictions from the same community (EDIT: Simon M lays out a case against extremizing, EDIT 2: I explain here a more robust way of choosing an extremizing factor).
- Lastly, it seems that empirically the weighting you use for the predictions matters much more than the aggregation method. I do not have yet great recommendations on how to do weighting, but it seems that weighting by recency of the prediction and by track record of the predictor works well at least in some cases.
There are reasons to believe I will have a better idea of which aggregation methods work best in a given context in a year. For example, it is not clear to me how to detect and deal with outliers, none of the current aggregation methods give consistent answers when annualizing probabilities and there is a huge unexplored space of aggregation functions that we might tap into with machine learning methods.
In conclusion and repeating myself: for the time being I would recommend people to stick to the geometric mean of odds as a default aggregate. I also encourage emphasizing the 10%, 50% and 90% percentile of predictions as well as the number of predictions to summarize the spread.
If you have a good pulse on a problem with the data, above I suggest some solutions you can try. But beware applying them blindly and choosing the outcome you like best.
Thanks to Simon M for his analysis of aggregation and weighting methods on Metaculus data. I thank Eric Neyman, Ozzie Gooen and Peter Wildeford for discussion and feedback.
(*) (Seaver, 1978) also performs different experiments comparing different pooling methods, and founds similar performance between the mean of probabilities and geometric mean of odds. However I suspect this is because the aggregated probabilities in the experiments were in a range where both methods give similar results.