The rewarding-more-active-forecasters problem seems severe and I'm surprised it's not getting more attention. If Alice and Bob both forecast the result of an election, but Alice updates her forecast every day (based on the latest polls) while Bob only updates his forecast every month, it doesn't make sense to compare their average daily Brier score.
Aha, of the top of my head one might go in the directions of (a) TD-learning type of reward; (b) variance reduction for policy evaluation.
After thinking for a few more minutes, it seems that forecasting more often but at random moments shouldn't impact the expected Brier score. But in practice people frequent forecasters are evaluated with respect to a different distribution (which favors information gain/"something relevant just happen") — so maybe some sort of importance sampling might help to equalize these two groups?
After thinking for a few more minutes, it seems that forecasting more often but at random moments shouldn't impact the expected Brier score.
In my toy example (where the forecasting moments are predetermined), Alice's Brier score for day X will be based on a"fresh" prediction made on that day (perhaps influenced by a new surprising poll result), while Bob's Brier score for that day may be based on a prediction he made 3 weeks earlier (not taking into account the new poll result). So we should expect that the average daily Brier score will be affected by the forecasting frequency (even if the forecasting moments are uniformly sampled).
In this toy example the best solution seems to be using the average Brier score over the set of days in which both Alice and Bob made a forecast. If in practice this tends to leave us with too few data points, a more sophisticated solution is called for. (Maybe partitioning days into bins and sampling a random forecast from each bin? [EDIT: this mechanism can be gamed.])
The long-term solution here is to allow forecasters to predict functions rather than just static values. This solves problems of things like people needing to update for time left.
In terms of the specific example though, I think if a significant new poll comes out and Alice updates and Bob doesn't, Alice is a better forecaster and deserves more reward than Bob.
The long-term solution here is to allow forecasters to predict functions rather than just static values. This solves problems of things like people needing to update for time left.
Do these functions map events to conditional probabilities? (I.e. mapping an event to the probability of something conditioned on that event happening)? How will this look like for the example of forecasting an election result?
In terms of the specific example though, I think if a significant new poll comes out and Alice updates and Bob doesn't, Alice is a better forecaster and deserves more reward than Bob.
Suppose Alice encountered the important poll result because she was looking for it (as part of her effort to come up with a new forecast). At the end of the day what we really care about is how much weight we should place on any given forecast made by Alice/Bob. We don't directly care about the average daily Brier score (which may be affected by the forecasting frequency). [EDIT: this isn't true if the forecasting platform and the forecasters' incentives are the same when we evaluate the forecasters and when we ask the questions we care about.]
Suppose Alice encountered the important poll result because she was looking for it (as part of her effort to come up with a new forecast).
This makes Alice a better forecaster, at least if the primary metric is accuracy. (If the metric includes other factors like efficiency, then we need to know eg. how many more minutes, if any, Alice spends than Bob).
At the end of the day what we really care about is how much weight we should place on any given forecast made by Alice/Bob.
If Alice updates daily and Bob updates once a month, and Alice has a lower average daily Brier score, then all else being equal, if you saw their forecasts at a random day, you should trust Alice's forecasts more*.
If you happen to see their forecasts on the day Bob updates, I agree this is a harder comparison, but I also don't think this is an unusually common use case.
I think part of the thing driving our intuition differences here is that I think lack of concurrency of forecasts (timeliness of opinions) is often a serious problem "in real life," rather than just an artifact of the platforms. In other words, you are imagining that whether to trust Alice at time t vs Bob at time t-1 is an unfortunate side effect of forecasting platforms, and "in real life" you generally have access to concurrent predictions by Alice and Bob. Whereas I think the timeliness tradeoff is a serious problem in most attempts to get accurate answers.
If you're trying to decide whether eg, a novel disease is airborne, you might have the choice of a meta-analysis from several months back, an expert opinion from 2 weeks ago, a prediction market median that was closed last week, or a single forecaster's opinion today.
___
Griping aside, I agree that there are situations where you do want to know "conditional upon two people making a forecast at the same time, whose forecasts do I trust more?" There are different proposed and implemented approaches around this, for example prediction markets implicitly get around this problem since the only people trading are people who implicitly believe that their forecasts are current, so the latest trades reflect the most accurate market beliefs, etc. (though markets have other problems like greater fool, especially since the existing prediction markets are much smaller than other markets).
*I've noticed this in myself. I used to update my Metaculus forecasts several times a week, and climbed the leaderboard fairly quickly in March and April. I've since slowed down to averaging an update once 3-6 weeks for most questions (except for a few "hot" ones or ones I'm unusually interested in). My score has slipped as a result. On the one hand I think this is a bit unfair since I feel like there's an important "meta" sense in which I've gotten better (more intuitive sense of probability, more acquired subject matter knowledge on the questions I'm forecasting). On the other, I think there's a very real sense that alex alludes to in which LinchSeptember is just a worse object-level forecaster than LinchApril, even if in some important meta-level ones (I like to imagine) I've gotten better.
This makes Alice a better forecaster
As long as we keep asking Alice and Bob questions via the same platform, and their incentives don't change, I agree. But if we now need to decide whether to hire Alice and/or Bob to do some forecasting for us, comparing their average daily Brier score is problematic. If Bob just wasn't motivated enough to update his forecast every day like Alice did, his lack of motivation can be fixed by paying him.
Here is a sketch of a formal argument, which will show that freshness doesn't matter much.
Let's calculate the average Brier score of a forecaster. We can see the contribution of hypothetical forecasts on day toward sum: . If forecasts are sufficiently random the expected number of days forecasts are active should be equal. Because , expected average Brier score is equal to the average of Briers scores for all days.
I'm also not sure I follow your exact argument here. But frequency clearly matters whenever the forecast is essentially resolved before the official resolution date, or when the best forecast based on evidence at time t behaves monotonically (think of questions of the type "will event Event x that (approximately) has a small fixed probability of happening each day happen before day y?", where each day passing without x happening should reduce your credence).
I mildly disagree. I think intuition to use here is that the sample mean is an unbiased estimator of expectation (this doesn't depend on frequency/number of samples). One complication here is that we are weighing samples potentially unequally, but if we expect each forecast to be active for an equal number of days this doesn't matter.
ETA: I think the assumption of "forecasts have an equal expected number of active days" breaks around the closing date, which impacts things in the monotonical example (this effect is linear in the expected number of active days and could be quite big in extremes).
I'm afraid I'm also not following. Take an extreme case (which is not that extreme given I think 'average number of forecasts per forecaster per question on GJO is 1.something). Alice predicts a year out P(X) = 0.2 and never touches her forecast again, whilst Bob predicts P(X) = 0.3, but decrements proportionately as time elapses. Say X doesn't happen (and say the right ex ante probability a year out was indeed 0.2). Although Alice > Bob on the initial forecast (and so if we just scored that day she would be better), if we carry forward Bob overtakes her overall [I haven't checked the maths for this example, but we can tweak initial forecasts so he does].
As time elapses, Alice's forecast steadily diverges from the 'true' ex ante likelihood, whilst Bob's converges to it. A similar story applies if new evidence emerges which dramatically changes the probability, if Bob updates on it and Alice doesn't. This seems roughly consonant with things like the stock-market - trading off month (or more) old prices rather than current prices seems unlikely to go well.
Thanks, everyone, for engaging with me. I will summarize my thoughts and would likely not actively comment here anymore:
On a meta-level, I am
This example is somewhat flawed (because forecasting only once breaks the assumption I am making) but might challenge your intuitions a bit :)
I didn't follow that last sentence.
Notice that in the limit it's obvious we should expect the forecasting frequency to affect the average daily Brier score: Suppose Alice makes a new forecast every day while Bob only makes a single forecast (which is equivalent to him making an initial forecast and then blindly making the same forecast every day until the question closes).
re: limit — a nice example. Please notice, that Bob makes a forecast on a (uniformly) random day, so when you take an expectation over the days he is making forecasts on you get the average of scores for all days as if he forecasted every day.
Let be the number of total days, be the probability Bob forecasted on a day , be the brier score of the forecast made on day :
I am a bit surprised that it worked out here because it breaks the assumption of the equality of the expected number of days forecast will be active. Lack of this assumption will play out if when aggregating over multiple questions [weighted by the number of active days]. Still, I hope this example gives helpful intuitions
.
Thanks for the explanation!
I don't think this formal argument conflicts with the claim that we should expect the forecasting frequency to affect the average daily Brier score. In the example that Flodorner gave where the forecast is essentially resolved before the official resolution date, Alice will have perfect daily Brier scores: , for any , while in those days Bob will have imperfect Brier scores: .
Thanks for challenging me :) I wrote my takes after this discussion above.
Do you have a source for the "carrying forward" on gjopen? I usually don't take the time to update my forecasts if I don't think I'd be able to beat the current median but might want to adjust my strategy in light of this.
Also, because the Median score is the median of all Brier scores (and not Brier score of the median forecast) it might still be good for your Accuracy score to forecast something close to community's median.
https://www.gjopen.com/faq says:
To determine your accuracy over the lifetime of a question, we calculate a Brier score for every day on which you had an active forecast, then take the average of those daily Brier scores and report it on your profile page. On days before you make your first forecast on a question, you do not receive a Brier score. Once you make a forecast on a question, we carry that forecast forward each day until you update it by submitting a new forecast.
I guess you're right (I read this before and interpreted "active foreast" as "forecast made very recently").
If they also used this way of scoring things for the results in Superforecasting, this seems like an important caveat for forecasting advice that is derived from the book: For example the efficacy of updating your beliefs might mostly be explained by this. I previously thought that the results meant that a person who forecasts a question daily will make better forecasts on sundays than a person who only forecasts on sundays.
Briefly:
I'm a fan of cultivating better forecasting ability in general and in the EA community in particular. Perhaps one can break down the benefits this way.
Training: Predicting how the future will go seems handy for intervening upon it to make it go better. Given forecasting seems to be a skill that can improve with practice, practising it could be a worthwhile activity.
Community accuracy: History augurs poorly for those who claim to know the future. Although even unpracticed forecasters typically beat chance, they tend inaccurate and overconfident. My understanding (tendentiously argued elsewhere) is taking aggregates of these forecasts - ditto all other beliefs we have (ibid) - allows us to fare better than we would each out on our own. Forecasting platforms are one useful way to coordinate in such an exercise, and so participation supplies a common epistemic good.[1] Although this good is undersupplied through the intellectual terrain, it may be particularly valuable for 'in house' topics of the EA community, as few outside it may contemplate these topics.
Self-knowledge/'calibration': Knowing one's ability as a forecast be useful piece of self-knowledge. It can inform how heavily we should weigh our own judgement in those rare cases where our opinion comprises a non-trivial proportion of the opinions we are modestly aggregating (ibid ad nauseum). Sometimes others ask us for forecasts, often under the guise of advice (I have been doing quite a lot of this with ongoing COVID-19 pandemic): our accuracy (absolute or relative) would be useful to provide alongside our forecast, so our advice can be weighed appropriately by its recipient.
Epistemic peer evaluation: It has been known for some to offer their opinion despite their counsel not being invited. In such cases, public disagreement can result. We may be more accurate in adjudicating these disagreements by weighing the epistemic virtue of the opposing 'camps' instead of the balance of argument as it appears to us (ibid - peccavi).
Alas, direct measures of epistemic accuracy can be elusive: people are apt to better remember (and report) their successes over their failures, and track records from things like prop betting or publicly registered predictions tend low-resolution. Other available proxy measures for 'intellectual clout' - subject matter expertise, social status, a writing style suffused with fulminant candenzas of melismatic and mellifluous (yet apropos and adroit) limerence of language - are inaccurate. Forecasting platforms allow people to publicly demonstrate their good judgement, and paying greater attention to these track records likely improves whatever rubbish approach is the status quo for judging others' judgement.
The latter two objectives require some means of comparing forecasters to one another.[2] This evaluation is tricky for a few reasons:
1. Metrics which allow good inter-individual comparison can interfere with the first two objectives, alongside other costs.
2. Probably in principle (and certainly in practice) natural metrics for this introduce various distortions.
3. (In consequence, said metrics are extremely gameable and vulnerable to Goodhart's law).
Suppose every EA started predicting on a platform like Metacalus. Also, suppose there was a credible means to rank all of them by their performance (more later). Finally, suppose this 'Metaculus rank' became an important metric used in mutual evaluation.
Although it goes without saying effective altruists almost perfectly act to further the common good, all-but-unalloyed with any notion of self-regard, insofar as this collective virtue is not adamantine, perverse incentives arise. Such as:
Places like GJO and Metaculus are aware of these problems, and so do not reward relative accuracy alone, either through separate metrics (badges for giving your rationale, 'upvotes' on comments, etc.) or making their 'ranking' measures composite metrics of accuracy and other things like activity (more later).[3]
These composite metrics are often better. Alice, who starts off a poor forecaster but through diligent practice becomes a good (but not great) and prolific contributor to a prediction platform has done something more valuable and praiseworthy than Bob, who was naturally brilliant but only stuck around long enough to demonstrate a track record to substantiate his boasting. Yet, as above, sometimes we really do (and really should) care about relative accuracy alone, and would value Bob's judgement over Alice's.
Even if we ignore the above, constructing a good metric of relative accuracy is much easier said than done. Even if we want to (as Tetlock recommends) 'keep score' of our performance, essentially all means of keeping score either introduce distortions, are easy to Goodhart, or are uninterpretable. To illustrate, I'll use all the metrics available for participating in Good Judgement Open as examples (I'm not on Metaculus, but I believe similar things apply).
Incomplete evaluation and strategic overconfidence: Some measures are only reported for single questions or a small set of questions ('forecast challenges' in the GJO). The challenges are those of variance and inadvertently rewarding overconfidence. 'Best performers' for a single question are invariably overconfident (and typically inaccurate) forecasters who maxed out their score by betting 0/100% the day a question opens and got lucky.
Sets of questions do a bit better (good forecasters tend to find themselves frequently at the top of the leaderboard), but their small number still allows a lot of volatility. My percentile across sets varies from top 0.1% to 60th or so. The former was on a set where I was on the 'right' side of the crowd for all of the dozen of the questions. Yet for many of these I was at something like 20% whilst the crowd was at 40% - even presuming I had edge rather than overconfidence, I got lucky that none of these happened. Contrariwise, being (rightly) less highly confident than the crowd will pay out in the long run, but the modal result in a small question-set is getting punished. The latter was a set where I 'beat the crowd' on most of the low probabilities, but tanked on an intermediate probability one - Brier scoring and non-normalized adding of absolute difference means this question explained most of the variance in performance across the set.[4]
If one kept score by ones 'best rankings', ones number of 'top ten finishes', or similar, this measure would reward overconfidence, as although this costs you in the long run, over the short run it can amplify good fortune.
Activity loading: The leaderboard for challenges isn't ranked by brier score (more later), but accuracy, essentially your Brier - crowd Brier. GJO evaluates each day, and 'carries forward' forecasts made before (i.e. if you say 52% on monday, you are counted as forecasting 52% on Tuesday, Wednesday, and every day until the question closes unless you change it). Thus - if you are beating the crowd - your 'accuracy score' is also partly an activity score, answering all the questions, having active forecasts as soon as they open all improve ones rank without being measures of good judgement.[5]
Metaculus ranks all users by a point score which (like this forum's karma system) rewards a history of activity rather than 'present performance': even if Alice was more accurate than all current metaculus users, if she joined today it would take her a very long time to overtake them.
Raw scores are meaningless without a question set: Happily, GJO uses a fairly pure 'performance metric' front and centre: Brier score across all of your forecasts.
Raw Brier Score:
[1]: Aside: one skill in forecasting which I think is neglected is formulating good questions. Typically our convictions are vague gestalts rather than particular crisp propositions. Finding useful 'proxy propositions' which usefully inform these broader convictions is an under-appreciated art.
[2]: Relative performance is a useful benchmark for self-knowledge as well as peer evaluation. Raw measures of absolute accuracy tend uninformative (much more later). If Alice tends worse than the average person at forecasting, she would be wise to be upfront about this lest her 'all things considered' judgements inadvertently lead others (who will typically aggregate better) astray.
[3]: I imagine it is also why the GJP doesn't spell out exactly how it selects the best GJO users for superforecaster selection.
[4]: One can argue the toss about whether there are easy improvements. One could make a scoring rule more sensitive to accuracy on rare events (Brier is infamously insensitive), or do some intra-question normalisation of accuracy. The downside would be this is intensely gameable for small question sets encouraging a 'pick the change up in front of the steamroller' strategy - overconfidently predicting rare events definitely won't happen will typically net one a lot of points, with the occasional massive bust.
[5] The 'carry forward' feature also means there are other ways to improve ones score which is more 'grinding' than 'talent', such as regularly reducing a forecast for an event happening in a time period as it begins to elapse. These are pretty minor though.
Great post! As you allude to, I'm increasingly of the opinion that the best way to evaluate forecaster performance is via how much respect other forecasters give them. This has a number of problems:
However, as you mention, other metrics have as many if not more problems. So on balance, I think as of 2020, the metric "who do other forecasters respect" currently carries more signal than any other metric I'm aware of.
That said, part of me still holds out hope that "as of 2020" is doing most of the work here. Forecasting in many ways seems to me like a nascent and preparadigm field, and it would not shock me if in 5-15 years we have much better ontologies/tools of measurement so that (as with other more mature fields) more quantified metrics will be better in the domain of forecaster evaluation than loose subjective human impressions.
I think this is an underrated point. Debating praiseworthiness seems like it can get political real fast, but I want to emphasize the point about value: there are different reasons you may care about participation in a forecasting platform, for example:
For the latter use case, I think people who participate regularly on the forecasting platforms, contribute a lot of comments, etc, usually improve group epistemics much more than people who are unerringly accurate on just a few questions.
Metaculus, as you mention, is aware of this, and (relative to GJO) rewards activity more than accuracy. I think this has large costs (in particular I think it makes the leaderboard a worse signal for accuracy), but is still on balance better.
__
A side note about Goodhart's law: I directionally agree with you, but I think Goodhart's law (related: optimizer's curse, specification gaming) is a serious issue to be aware of, but (as with nuance) I worry that in EA discussions about Goodhart's law there's a risk of being "too clever." At any point you're trying to collapse the complex/subtle/multivariate/multidimensional nature of reality to a small set of easily measurable/quantifiable dimensions (sometimes just one), you end up losing information. You hope that none of the information you lose is particularly important, but in practice this is rarely true.
Nonetheless, it is the case that to (a first approximation), imperfect metrics often work in getting the things you want to get done. For example, the image/speech recognition benchmarks often have glaring robustness holes that are easy to point out, yet I think it's relatively uncontroversial that in many practical use cases, there are a plethora of situations where ML perception classifiers, created in large part by academics and industry optimizing along those metrics, are currently at or will soon approach superhuman quality.
Likewise, in many businesses, a common partial solution for principle-agent problems is for managers to give employees metrics of success (usually gameable ones that are only moderately correlated with the eventual goal of profit maximization). This can result in wasted effort via specification gaming, but nonetheless many businesses still end up being profitable as a direct result of employees having direct targets.
I think (as with some of our other "disagreements") I am again violently agreeing with you. Your position seems to be "we should take metrics as useful indicators but we should be worried about taking them too seriously" whereas my position is closer to "we should be worried about taking metrics too seriously, but we should care a lot about the good metrics, and in the absence of good metrics, try really hard to find better ones."