The real mean of war casualties calculated by the dual method is considerably larger than the sample mean.

Judge as of 2015-05-18 against the paper Taleb co-authored with Cirillo ("On the statistical properties and tail risk of violent conflicts" / the dual-distribution method). True: the paper's dual (or 'shadow') mean of war casualties, computed with its stated method, is substantially above the naive sample mean, e.g. clearly more than a few percent and described or shown as a material gap. Mostly true: the dual mean is higher and the gap is material, but the claim gets a minor detail wrong, such as the exact size of the gap or which dataset was used. Mixed: the dual mean is higher in some specifications or datasets but not in others, or only marginally higher overall. Mostly false: the dual mean is only trivially higher (within a few percent), so 'considerably' overstates a real but small difference. False: the paper's dual mean is equal to or lower than the sample mean.

Set before the research began.

1. E1 Violent Warfare: On the Wane? nassimtaleb.org (republishing Mark Buchanan's Medium article) · 18 May 2015 Own record

   “In actual numbers, Cirillo and Taleb find that the true expected mean — say, the number of deaths we should expect over the next half century — is actually about three times higher than what we’ve seen in the past.”

2. E2 Violent Warfare: On the Wane? nassimtaleb.org (republishing Mark Buchanan's Medium article) · 18 May 2015 Own record

   “Related to the above, it also turns out to be >90% certain that the true mean of the process is higher than the observed mean.”

3. E3 Letters/Puzzle Significance (Royal Statistical Society / Oxford Academic) · 6 June 2016

   “In “What are the chances of war?” (page 44, April 2016) Pasquale Cirillo and Nassim Taleb attempt to educate us on fat-tailed distributions, but they need not have bothered.”

4. E4 On the shadow moments of apparently infinite-mean phenomena ar5iv (arXiv 1510.06731) Own record

   “War casualties, operational risk, environment blight, complex networks and many other econophysics phenomena are possible fields of application.”

5. E5 On the shadow moments of apparently infinite-mean phenomena ar5iv (arXiv 1510.06731) Own record

   “This also implies that the sample mean is essentially useless for making inference, in addition to any considerations about robustness [6].”

6. E6 On the shadow moments of apparently infinite-mean phenomena ar5iv (arXiv 1510.06731) Own record

   “It is worth noticing that if \(L^{\ast}=L\), then \(E[Y|Y>L^{\ast}]=E[Y]\), i.e. the conditional mean of \(Y\) above \(Y\) is exactly the mean of \(Y\).”

7. E7 Stochastic Tail Exponent For Asymmetric Power Laws arXiv (1609.02369v3) Own record

   “We also consider the situation of capped power laws (i.e. with compact support), and apply it to the study of violence by Cirillo and Taleb (2016). We show that uncertainty concerning the historical data increases the true mean.”

8. E8 Letters/Puzzle Significance (Oxford Academic, Royal Statistical Society) · 6 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.”

9. E9 The war file lucid.rodeo ("incerto" section; site owner not shown on the page) · 1 January 2016 Own record

   “Their bounded-support (dual) transform recovers a finite shadow mean — about 3.69×10⁷ casualties for rescaled data at the 145k threshold, against a sample mean of 1.77×10⁷”

10. E10 New Study: The world's a lot more violent than reported History News Network · 30 May 2015

    “the risk of violent conflict has not been decreasing, but is rather underestimated by techniques relying on naive year-on-year changes in the mean, or using sample mean as an estimator of the true mean of an extremely fat-tailed phenomenon;”

11. E11 Moments of undersampled distributions: Application to the size of epidemics arXiv (author: Álvaro Corral, Centre de Recerca Matemàtica, Barcelona; Universitat Autònoma de Barcelona; Complexity Science Hub Vienna) · March 2024

    “Here, we (i) show that such a transformation is ad hoc and unphysical; (ii) propose alternative transformations and distributions (also ad hoc)”

A summary of the Cirillo–Taleb analysis reports a dual (shadow) mean of about 3.69×10⁷ casualties against a sample mean of 1.77×10⁷, roughly double (E9). Mark Buchanan's write-up from the claim date says the true expected mean is about three times the past observed level and more than 90% likely to exceed it (E1, E2). History News Network independently reports that the paper finds the sample mean underestimates the true mean (E10). The two sources that give figures, neither of them independent of the subject, differ on the ratio (roughly double in E9, about three times in E1), and both show a large gap; the other sources describe the true mean as higher, or the sample mean as an underestimate, without giving its size [E2][E10].