The Unholy Methods of Soothsayers

Defensive Forecasting and the mechanical crank of prediction.

Hi there, argmin readers! Today’s post is a live blog of Class 18 of my graduate seminar “Forecasting: A Critical Retrospective.” In this class, we covered one of my favorite topics: Defensive Forecasting. The syllabus and list of past posts are here.

“A good Christian must beware of mathematicians and those soothsayers who make predictions by unholy methods, especially when their predictions come true, lest they ensnare the soul through association with demons.” - Augustine of Hippo.

We’ve spent the semester examining application areas with strong cultures of forecasting. But one of the central reasons I proposed this class in the first place was to examine the culture of forecasting itself. American society is obsessed with applying forecasting, prediction markets, and gambling to… everything. We’ll spend the remaining classes investigating this oppressive culture.

Next class we’ll discuss prediction markets, and in the following we’ll get to that whole p(doom) thing. But to talk about both, we first have to step back and return to a logical fallacy of forecasting raised in the first lecture of the class and by Saint Augustine millennia ago.

Certainly, we believe that if you have a good understanding of a system, you should be able to predict its future trajectory on some reasonable time horizon. We can quibble about how long such a projection would be, but understanding implies some ability to talk through potential future outcomes and their likelihoods. But what about the converse? If you can predict the future, does that mean you understand the world? The answer is no, and you can mathematically prove it.

Superforecasters themselves say you can train yourself to be better at forecasting. This training requires no expert domain knowledge. They advise you to find the best set of past events that match the thing you are trying to predict (i.e., find the reference class). You are then told to compute the base rate in that reference class. Then, you are supposed to nudge the base rate based on whether new information makes the event more or less likely than in the past. This last step is the least well-specified, but you guide yourself by making sure that your track record remains well-calibrated. Following these rules, good forecasting is and should be mostly algorithmic bookkeeping devoid of domain knowledge. You could certainly write computer code to do it. You don’t even need an LLM. A good forecaster is a mechanical pattern recognizer.

Mechanical pattern recognition is great because it doesn’t require understanding. This is why machine learning is such a powerful technology.

You can make this very formal using the framework of Defensive Forecasting, a term coined by Vladimir Vovk, Akimichi Takemura, and Glenn Shafer to describe a class of algorithms that mechanize the meta-forecasting approach I describe above. This work has connections to paradoxes in calibration raised by Foster and Vohra in the 90s, who showed you could produce well-calibrated predictions knowing nothing about data-generating processes. It is closely related to Blackwell Approachability, where we aim to adaptively steer multiple objectives at once. Blackwell approachability is also equivalent to no-regret learning, so our friend online gradient descent can be interpreted as clever bookkeeping. Defensive forecasting is also equivalent to recently developed frameworks for multicalibration, and hence subsumes the problem of outcome indistinguishability.

Juanky Perdomo and I wrote what we hope is an accessible introduction to this topic, laying out all of these connections in as elementary a way as we could. One simple idea makes the whole thing work.

That idea is perhaps easiest to describe in the language of outcome indistinguishability. We know from the law of large numbers that the variance of a bunch of iid draws from a Bernoulli distribution shrinks at a rate of one divided by the number of draws. If a set of outcomes tracks a set of predictions at this rate, then it’s impossible to say the outcomes weren’t sampled from a probability distribution defined by those probabilistic predictions. Defensive forecasting exploits this indistinguishability. It turns out you can always pick a prediction so that it looks like the law of large numbers held. Moreover, you can find this prediction by correcting errors that pulled you away from the law of large numbers in the past.

This is possible because the evaluation rules are proposed in advance. These rules can always be reduced to some sort of moment-matching problem (i.e., a multicalibration problem), and moment matching with defensive forecasting can be derived using elementary algebra. For a simple example of this reduction, check out this blog post from last year. For the full details of the reduction, see Section 5 of my paper with Juanky linked above.

The metrics we use to study prediction fidelity of uncertain events are all gameable in the sense that they can be maximized without a “world model.”¹ These metrics include calibration checks and proper scoring rules. Forecasting is not about understanding. It’s about meeting expectations in an evaluation game. Once the rules are set, the method is determined. Gather as many covariates as you can, write down the appropriate moments to match, and then just let the machine sort it out.

There is no better illustration of metrical determinism than forecasting. The question and evaluation wholly determine the forecast. And that’s where we should focus our inquiry: why are we forecasting these things? And how are we judging forecasts? Once we decide what we’re forecasting, the best forecasting strategy is predetermined. We can’t know the future, but we can mechanically generate forecasts that look like the future came from our model. This seductive trap makes us think forecasters know something we don’t. Heed Saint Augustine’s advice and don’t fall for it.

1. Use whatever definition you want for that buzzword.