The dual method allows calculation of the real mean of war casualties.

Judged as of 2015-05-18. True: the published 'dual distribution' method (Cirillo and Taleb, 2015–16) does what the claim says: it gives a mathematically valid estimate of the true (shadow) mean of war casualties, taking the bounded maximum into account, and the scholarly literature accepts it as doing so. Mostly true: the method does give such an estimate, but only under stated assumptions (e.g. a fitted tail, an upper bound), so 'real mean' slightly overstates it as an estimate. Mixed: the method's validity is substantially disputed in the literature, with roughly balanced support. Mostly false: the method yields only a model-dependent estimate that published critiques show to be unreliable, so calling it the 'real' mean overstates an unproven result. False: the method is shown to be mathematically invalid for estimating the mean.

Set before the research began.

1. E1 Moments of undersampled distributions: Application to the size of epidemics arXiv · 31 January 2026

   “Cirillo and Taleb [Nature Phys. 16, 606 (2020)] propose a transformation from a so-called dual variable, which displays a power-law tail, to the total number of fatalities, which becomes bounded by the total world population. Here, we (i) show that such a transformation is ad hoc and unphysical;”

2. E2 Moments of undersampled distributions: Application to the size of epidemics arXiv · 31 January 2026

   “the real problem is not the non-existence of the mean value for power-law tailed distributions but the fact that the tail of the different theoretical distributions (which is what distinguishes one model from the other) is far from being well sampled with the available number of empirical data.”

3. E3 EMPTY STATISTICS quixoticfinance.com (blog) · 29 December 2015

   “(briefly, it allows them to introduce a ‘novel approach to apply extreme value theory’, the dual distribution).”

4. E4 What are the chances of war? TU Delft Research Portal (Significance, Wiley) · 8 April 2016 Own record

   “Pasquale Cirillo and Nassim Nicholas Taleb respond to December's “Ask a Statistician” column, explaining more about their work to understand the risk of violent conflicts”

5. E5 Moments of undersampled distributions: Application to the size of epidemics arXiv · 31 January 2026

   “Here, we (i) show that such a transformation is ad hoc and unphysical;”

6. E6 Moments of undersampled distributions: Application to the size of epidemics arXiv · 31 January 2026

   “We demonstrate that this is due to an unphysical divergence of the probability density of the number of fatalities at the highest-possible value.”

7. E7 Moments of undersampled distributions: Application to the size of epidemics arXiv · 31 January 2026

   “The lesson learned from here is that, as moments cannot be constrained, it is pointless to try to calculate them and one should rely on other metrics to assess risk.”

8. E8 Letters/Puzzle Significance (Royal Statistical Society / Oxford Academic) · 1 June 2016

   “Contrary to Cirillo and Taleb, we are perfectly aware that a stretch of time without a big war does not imply that a big war cannot happen.”

Among the evidence gathered, the only direct critique of the dual-distribution method is one arXiv preprint (E1, E2, E5, E6, E7). It calls the Cirillo–Taleb transformation ad hoc and unphysical and argues that moments cannot be constrained. However, it addresses the epidemic-fatality application rather than war casualties, it is a single preprint, and it postdates the 2015 judgment date. E12 disputes a different inference, not the mean calculation, and E3 and E4 only describe the method. This evidence does not show whether the scholarly literature accepts, disputes or rejects the method as a valid estimator of the war-casualty mean, so the criteria cannot be applied with confidence.