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Dear Dr. Widmann,

Thanks for your interesting work titled "Calibration tests in multi-class classification: A unifying framework." I have a question regarding the derivation of equivalence, specifically in Lemma E.2 (Explicit formulation) of your appendix E.

In this lemma, it is stated that SKCE[k,g] = E[<∆, k(G,G')∆'>], where ∆ = E[e|G] − G = E[e − G|G]. By substituting ∆, I obtained SKCE[k,g] = E[<E[e − G|G], k(G,G') E[e' − G'|G']>. However, I am unsure how the conditional expectation can be eliminated to yield SKCE[k,g] = E[<e − G, k(G,G') (e'−G')>].

Personally, I believe that estimating such conditional expectations is crucial, particularly in high-dimensional cases where it can be challenging. Therefore, I am curious to know the reason behind its removal. Your response to this simple question would be greatly appreciated.

Thank you for your time and consideration.