5. Supplementary Methods#
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5.1. Transitivity of BEMD comparisons#
Given three independent continuous random variables \(R_A\), \(R_B\) and \(R_C\), define the probabilities
For some threshold \(\falsifythreshold\), we would like these to satisfy a transitivity relation of the form
as this would reduce the required number of pairwise comparisons between models (see section Model discrepancy as a baseline for non-stationary replications, Table 2.1 and Eq. 2.13 in the main text).
It is known that Eq. 5.1 does not hold for \(\falsifythreshold = \tfrac{1}{2}\); classic counterexamples in this case are non-transitive dice [71]. However, the set of probabilities \(\mathcal{S} \coloneqq \{ B_{AB}, B_{BC}, B_{CA} \}\) does satisfy a property known as dice-transitivity [38, Theorem 19], from which one can derive that Eq. 5.1 holds when \(\falsifythreshold = \varphi^{-1}\), where \(\varphi\) is the golden ratio. This result appears as a comment below Theorem 3 in Baets and Meyer [39], but to our knowledge has otherwise remained unknown. We provide a short self-contained derivation below, for the convenience of the reader.
The definition of dice-transitivity is obtained by substituting equation (9) of De Schuymer et al. [38] into equation (6) of the same reference. For our purposes we are interested in the resulting upper bound
where \(α\), \(β\), and \(γ\) are respectively the lowest, middle and highest value of \(\mathcal{S}\). In other words, \(\{α,β,γ\} = \mathcal{S}\) and
Suppose that, as given in Eq. 2.13, we have
We wish to use Eq. 5.2 to establish an upper bound on \(B_{CA}\). We do not know a priori how the probabilities are ordered, so we consider the six possible cases:
The assumptions of Eq. 5.4 translate to \(α > \varphi^{-1}\) and \(β > \varphi^{-1}\). We seek a bound on \(γ\). Rearranging Eq. 5.2, we then have
which contradicts Eq. 5.3.
The argument is exactly analogous, except that we seek a bound on \(β\). We get
which again contradicts Eq. 5.3.
Therefore the only possible cases are \(βγα\) and \(γβα\), which means that \(B_{CA}\) must be the smallest of the three probabilities. These two final cases provide the upper bound on \(B_{CA}\):
Thus Eq. 5.1 holds for \(\falsifythreshold = \varphi^{-1}\). More generally, we have \(α = B_{CA}\) whenever both \(B_{AB}\) and \(B_{BC}\) are greater than \(\varphi^{-1}\). In this case Eq. 5.5 implies
and therefore
Eq. 2.13 which we gave in the main text is a special case of Eq. 5.6.