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Kindly comments on Raising the forecasting waterline (part 1) - Less Wrong
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How does GJP score predictions that change over time?
They compute your Brier score for each day that the question is open, according to what your forecast is on that day, and average over all days.
Suppose you start at 80%, six days pass, you switch to 40% three days before the deadline, and the event doesn't happen, your score is (6*(0.8)^2+3*(0.4)^2)/9 = .48, which is a so-so score - but an improvement over the .64 that you'd get if you didn't change your mind.