Taleb states that one can further verify the Brier score has thinner tails than the Gaussian, as its kurtosis is lower than 3.

This is a mathematical claim, judged against the distribution of the Brier score in the setting Taleb's 2019 paper specifies (binary forecasts with stated probability distributions). True: in that setting the Brier score's excess kurtosis is negative (kurtosis below 3), so its tails are thinner than a Gaussian's, and kurtosis below 3 properly supports or is consistent with that tail comparison. The paper presents the low kurtosis as verification, so 'because' is read as 'as shown by'. Mostly true: kurtosis is below 3 and the tails are thinner (for example, the score is bounded), but only for most of the paper's parameter values rather than all of them, or the link between kurtosis and tail thickness is stated loosely. Mixed: kurtosis is below 3 for some of the relevant parameter values but not for others, so the claim holds only partly. Mostly false: the tails are thinner than Gaussian but kurtosis is not generally below 3, so the stated verification fails. False: a derivation or simulation shows kurtosis is 3 or more in the stated setting and the tails are not thinner than Gaussian.

Set before the research began.

1. E1 On misconceptions about the Brier score in binary prediction models arXiv (HTML rendering)

   “The expectation of Brier score in (1.1) and its distribution depend on the true outcome probabilities \(q_{i}\), which are generally unknown, even when these are used as predictions.”

2. E2 On misconceptions about the Brier score in binary prediction models arXiv (2504.04906, clinical-statistics commentary)

   “On the other hand, when they are mostly concentrated around e.g. \(0.5\), the expectation of the Brier score (1.1) with perfect predictions is close to \(0.25\).”

E1 and E4 come from an independent commentary on the Brier score. They say only that the score's distribution and expected value depend on the true outcome probabilities. Neither reports a kurtosis value, a derivation or simulation of kurtosis, or any comparison of tail thickness with the Gaussian. The evidence does not settle whether the Brier score's kurtosis is below 3 in the setting of Taleb's 2019 paper.