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Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Rational predictions often update predictably, published by Gregory Lewis on May 15, 2022 on The Effective Altruism Forum. BLUF: One common supposition is a rational forecast, because it ‘prices in’ anticipated evidence, should follow a (symmetric) random walk. Thus one should not expect a predictable trend in rational forecasts (e.g. 10% day 1, 9% day 2, 8% day 3, etc.), nor commonly see this pattern when reviewing good forecasting platforms. Yet one does, and this is because the supposition is wrong: ‘pricing in’ and reflective equilibrium constraints only entail the present credence is the expected value of a future credence. Skew in the anticipated distribution can give rise to the commonly observed “steady pattern of updates in one direction”. Such skew is very common: it is typical in 'will event happen by date' questions, and one’s current belief often implies skew in the expected distribution of future credences. Thus predictable directions in updating are unreliable indicators of irrationality. Introduction Forecasting is common (although it should be commoner) and forecasts (whether our own or others) change with further reflection or new evidence. There are standard metrics to assess how good someone’s forecasting is, like accuracy and calibration. Another putative metric is something like ‘crowd anticipation’: if I predict P(X) = 0.8 when the consensus is P(X) = 0.6, but over time this consensus moves to P(X) = 0.8, regardless of how the question resolves, I might take this to be evidence I was ‘ahead of the curve’ in assessing the right probability which should have been believed given the evidence available. This leads to scepticism about the rationality of predictors which show a pattern of ‘steadily moving in one direction’ for a given question: e.g. P(X) = 0.6, then 0.63, 0.69, 0.72 . and then the question resolves affirmatively. Surely a more rational predictor, observing this pattern, would try and make forecasts an observer couldn’t reliably guess to be higher or lower in the future, so forecast values follow something like a (symmetrical) random walk. Yet forecast aggregates (and individual forecasters) commonly show these directional patterns if tracking a given question. Scott Alexander noted curiosity about this behaviour; Eliezer Yudkowsky has confidently asserted it is an indicator of sub-par Bayesian updating. Yet the forecasters regularly (and predictably) notching questions up or down as time passes are being rational. Alexander’s curiosity was satisfied by various comments on his piece, but I write here as this understanding may be tacit knowledge to regular forecasters, yet valuable to explain to a wider audience. Reflective equilibrium, expected value, and skew One of the typical arguments against steadily directional patterns is they suggest a violation of reflective equilibrium. In the same way if I say P(X) = 0.8, I should not expect to believe P(X) = 1 (i.e. it happened) more than 80% of the time, I shouldn’t expect to believe P(X) [later] > P(X) [now] more likely than not. If I did, surely I should start ‘pricing that in’ and updating my current forecast upwards. Market analogies are commonly appealed to in making this point: if we know the stock price of a company is more likely than not to go up, why haven’t we bid up the price already? This motivation is mistaken. Reflective equilibrium only demands one’s current forecast is the expected value of one’s future credence. So although the mean of P(X) [later] should equal P(X) [now], there are no other constraints on the distribution. If it is skewed, then you can have predictable update direction without irrationality: e.g. from P(X) = 0.8, I may think in a week I will - most of the time - have notched this forecast slightly upwards, but less of the time notching it further downwa...