In short: In light of a recent result by Eric Neyman, I tentatively recommend extremizing aggregated forecasts by a factor $d=\frac{n(\sqrt{3n^2-3n+1}-2)}{n^2-n-1}$, which is approximately $d\approx \sqrt{3}\approx 1.73$ for $n>50$ :

When historical resolution rates are available and reasonably constant through time, I more hesitantly recommend using the aggregate:

where $\log O_{\text{baseline}}$ are the historical logarithmic odds of a positive resolution.

Neyman's result is theoretical and it is not clearly applicable in this context, but I show it has good performance on Metaculus data.

The forecast aggregation war continues.

Extreme forecasting

Extremizing is the practice of adjusting aggregated forecasts towards a extreme - either 0% or 100% probability.

If the experts predictions are odds $o_1,\dots ,o_n$, then an extremized aggregation of the odds is:

for a given extremization factor $d$.

Extremizing is a common practice in academic contexts. It is usually justified on the grounds that different experts have access to different information (Baron et al, 2014). 

And indeed it has been shown to have a good performance in practice. (Satopää et al, 2014) showed that forecasts in a collection of geopolitical questions are optimally aggregated when using an extremizing factor $d\in [1.161,3.921]$.

However there are reasons to be skeptical of extremizing. It introduces a degree of freedom in the choice of the extremizing parameter. Results like Satopää's where an optimal extremizing factor is derived in hindsight risk overfitting this parameter.

Before making a blank recommendation of extremizing, we would prefer to have a grounded way of choosing an extremizing factor.

Why I became an extremist

Neyman and Roughgarden (2021) have taken a close look at the theory of forecast aggregation and extremizing [1]. 

They suggest and analyse an unconventional extremizing method, where they move the aggregate estimate away from its baseline value. Concretely, their estimate as applied to logodd aggregation would be:

which can be rearranged as

or as 

This is the same as classical extremizing when we assume $\log O_{\text{baseline}}=0$, though as we will see later it might be better to use a historical estimate.

Neyman and Roughgarden show that when using an extremizing factor $d=\frac{n(\sqrt{3n^2-3n+1}-2)}{n^2-n-1}$ this estimate outperforms (in a certain sense) the logodds average.

As $n$ grows, the recommended extremizing factor approaches $d\to \sqrt{3}\approx 1.73$. In practice, for $n>50$ the approximation $d\approx \sqrt{3}\approx 1.73$ is already pretty good [2].

In which sense does the extremized prediction perform better? The authors analysis is done in terms of what they call the approximation ratio. This measures how close the aggregated estimate approaches gets to the square loss of the idealized prediction of an expert with access to everyone's information [3]. 

The authors find that under the projective substitutes condition their aggregation scheme performs better than a simple average, in terms of the approximation ratio [4].

What is this projective substitutes condition? Essentially, it states that there are diminishing marginal returns to more forecasts [5]. I think that this is a plausible assumption in the context of forecast aggregation, though it is not a guarantee [6].

Does the recommended aggregation strategy perform well in practice? Yes it does.
I looked at 899 resolved binary questions in Metaculus, and compared several aggregation methods. The results of this analysis are included in the appendix.

In short, when assuming $\log O_{\text{baseline}}=0$, Neyman's extremizing factor outperforms in this dataset most other methods I tried, and it is on par with a optimized and constant extremizing rate and with the current Metaculus prediction method.

But that is not all. When I used $\log O_{\text{baseline}}$ equal to the resolution rate of the currently resolved binary questions in Metaculus the results significantly outperform all other methods I tried, including the current Metaculus prediction [7].

In conclusion

I harbored some doubts about extremizing. It was common practice and there was some empirical evidence in its favour. But I lacked a convincing argument to rule out overfitting as the reason for the increased performance.

This has now changed. In Neyman and Roughgarden (2021) I find both a sound argument in favour and a recipe to automatically choose an extremizing rate.

In response, I now tentatively recommend extremizing average log odds as a default method for aggregating, using Neyman's method to choose an extremizing factor [8].

I am more hesistant to recommend the more complex extremization method where we use the historical baseline resolution log-odds and aggregate forecasts as:

I think I would recommend this in cases where there is convincing evidence of a relatively constant resolution rate through time. For example, I believe this is the case for Metaculus binary questions. 

Note that a big part of my recommendation relies on my idiosincratic reading of Neyman and Roughgarden's results, the assumption that logodd forecast aggregation satisfies the projective substitutes condition and the empirical performance on Metaculus data. 

While my beliefs have changed enough to change my best-guess recommendation, I do not see the question as settled. Further theoretical and empirical evidence could easily change this conclusion, either showing how this result does not apply or coming up with a better result.

Acknowledgements

Many thanks to Eric Neyman for writing his paper and hepfully answering my questions about it.

I thank Simon M for the script and discussion of previous work, and Lawrence Phillips for reproducing the results and finding a mistake.

My work is financially supported by the Open Philanthropy Project. 

Appendix: Testing Neyman's method on Metaculus data

I used 899 resolved binary questions in Metaculus to study the empirical performance of Neyman's suggested method for choosing an extremizing factor.

In the table above I show the performance of several aggregation methods: the mean of logodds, the mean of probabilities, the median, the extremized average of logodds and Neyman's proposed aggregation method.

I include unweighted and weighted versions of each - for the weighted version we weight each experts prediction by its recency, following the same procedure as Metaculus.

To compute the Neyman aggregation, we use the formula:

where $d=\frac{n(\sqrt{3n^2-3n+1}-2)}{n^2-n-1}$, where $n$ is the number of respondents.

I used both an uninformative prior $p=0.5$ and the actual resolution rate among the questions ($p=0.3571$) to derive the baseline logodds $\log O_{\text{baseline}}$.

For the extremized average of logodds shown I chose the extremization factors $d$ that approximately minimize the log score.

I also include the score of the default metaculus aggregation and the baseline score we would have gotten with a constant prediction equal to the mean resolution of the questions ($p=35.71\%$). 

I included the Brier score and the log loss score - lower scores are better.

Note that the Neyman aggregation performed quite well. 

When assuming a zero baseline logodds it performs better than all simple methods. It also (barely) outperforms the Metaculus aggregation in terms of log score (though not in terms of Brier score).

When assuming a baseline logodds that match the actual empirical rate it outperforms all other methods I tried. This is slighly misleading, since we only have access to the empirical resolution rate in hindsight. It is still quite encouraging. 

The script to replicate my findings is here. It is based off Simon M's script here.

Footnotes

[1] Note that in the paper the aggregation is discussed in the context of estimating real-value quantities. Here I am independently arguing that their framework can reasonably apply to the context of estimating discrete logodds and plausibly continuous logdensities.

[2] Note that this factor $d\approx \sqrt{3}$ falls within the confidence interval $d\in [1.161,3.921]$ of optimal extremizing factors found by Satopää et al. 

[3] Suppose we are trying to forecast the odds $Y$ of an event. Each of $n$ experts is granted a piece of evidence $E_i$ that they use to elicit a forecast $Y_i:=Y|E_i$. We then summarize their beliefs with an aggregate forecast $\hat{Y}=f(Y_1,...,Y_n)$. Our goal is to compare how good this aggregate forecast $\hat{Y}$ is with respect to the forecast an expert would make, if they had access to all information, ie $Y_{\left[n\right]}:=Y|E_1,...,E_n$.

The approximation ratio of an aggregate estimator $\hat{Y}$ then would be:

Note that the approximation ratio satisfies that $AR(Y_{\left[n\right]},Y|E)=1$ and $AR\left(E\right[Y],Y|E)=0$.

[4] The relevant part is theorem 4.1.

We contrast this result with theorem 3.1, which shows the approximation ratio in the case of a simple average.

In short, they find that with their extremizing method they can achieve an approximation ratio of $\frac{3\sqrt{3}}{2n}-O\left(\frac{1}{n^2}\right)\approx 2.598/n$, while a non-extremized average attains an approximation ratio of $\frac{1+\sqrt{3}/2}{n}-O\left(\frac{1}{n^2}\right)\approx 1.866/n$  (the approximations hold for large $n$).

According to the authors analysis this is the best possible extremizing method of the form $f(Y_1,...,Y_n,E[Y\left]\right)$ with $f$ linear in terms of optimizing the approximation ratio. But better non-linear methods might exist, or other results for more appropriate measures of optimality.

[5] Being inexact, it means that the information gap between a group of experts sharing all their evidence and a strict subset of it would decrease if we were to add another expert to both the group and the subset.

[6] For example, in Ord's Jack, Queen and King example the projective substitutes condition does not hold, since the joint information of players A and B complements each other. The weaker weak substitutes condition does not hold either.

[7] Using the resolution baseline logodds is in a sense cheating, since I used the results of the questions to estimate this baseline. But this is an encouraging result assuming that the positive resolution rate (PRR) is roughly constant over time.

To investigate this, I ran a bootstrap study on the 995 resolved binary questions on Metaculus. I resampled with replacement 995 questions B=100,000 times and computed the PRR. The resulting 90% confidence interval is $[0.3315;0.3571;0.3838]$.

I also studied the rolling positive resolution rate. I computed the PRR up to the ith question for $1\le i\le 995$. The resulting 90% confidence interval is $[0.2754;0.3068;0.3729]$.

Both of these results are weak evidence that the PRR in Metaculus is relatively stable.

[8] Note that Eric Neyman himself raised a few concerns when I showed him a draft of the post:

1. It is not clear that the approximation ratio is the thing we care most about when aggregating forecasts. Neyman remarks that the KL divergence between the aggregation and the "true odds" would be a better metric to optimize.
2. It is not clear that "prior/ baseline log-odds" means in the contexts of aggregating forecasts. Recall that this method was developed to aggregate point estimates of real values. We are taking a license when assuming the analysis would apply to aggregating logodds as well.
3. Ultimately, he argues that the extremization factor to use should be empirically derived. These theoretical results provide guidance in choosing how to aggregate results, but its no substitute for empirical evidence.

I eagerly look forward to his future papers. 

Bibliography

Neyman, Eric, and Tim Roughgarden. 2021. ‘Are You Smarter Than a Random Expert? The Robust Aggregation of Substitutable Signals’. ArXiv:2111.03153 [Cs], November. http://arxiv.org/abs/2111.03153.

Satopää, Ville A., Jonathan Baron, Dean P. Foster, Barbara A. Mellers, Philip E. Tetlock, and Lyle H. Ungar. 2014. ‘Combining Multiple Probability Predictions Using a Simple Logit Model’. International Journal of Forecasting 30 (2): 344–56. https://doi.org/10.1016/j.ijforecast.2013.09.009.

Baron, Jonathan, Barb Mellers, Philip Tetlock, Eric Stone, and Lyle Ungar. 2014. ‘Two Reasons to Make Aggregated Probability Forecasts More Extreme’. Decision Analysis 11 (June): 133–45. https://doi.org/10.1287/deca.2014.0293.

Sevilla, Jaime. 2021. ‘My Current Best Guess on How to Aggregate Forecasts’. EA Forum, 6 October 2021. https://forum.effectivealtruism.org/posts/acREnv2Z5h4Fr5NWz/my-current-best-guess-on-how-to-aggregate-forecasts.

Sevilla, Jaime. 2021. ‘When Pooling Forecasts, Use the Geometric Mean of Odds’. EA Forum. 3 September 2021. https://forum.effectivealtruism.org/posts/sMjcjnnpoAQCcedL2/when-pooling-forecasts-use-the-geometric-mean-of-odds.