van Rooyen, B. (2025). On Experiments. arXiv:2508.08288.
Key takeaways
From the reading, NOTE-1430 (Skimmed: §4 not read).
- Fixed-prior randomization theorem (Theorem 26, pp. 18–19). For a fixed prior π and finite sets, Risk^π_L(e) ≤ Risk^π_L(e′) + ε‖L‖∞ for all action sets and losses if and only if min_f E_{θ∼π} V(f∘e(θ), e′(θ)) ≤ ε. The proof of the forward implication fixes a rule d′ for e′, applies the minimax theorem to the risk gap as a function of the loss and of e’s rule, and takes the action set to be e′‘s observations and d′ the identity.
- Its limits. It is stated for a fixed prior, with the deficiency an average over θ, not a maximum. Its reverse implication is right for losses in [0, 1] and off by a factor of 2 for signed losses, since V is half the ℓ1 distance.
- Data processing as a functor (Theorem 24). Any risk minimised over the strategies an experiment can reach cannot fall under garbling.
- Deficiency by linear programme (Lemma 30). The programme as printed sums the absolute deviations, so its value is 2ξ
Standing in the record
Filed on 2026-10-30 for CLAIM-488. Theorem 26 is the converse that claim waits on, proved for a fixed prior. The uniform version that the claim’s first step needs, deficiency as a maximum over the unknown, needs one more minimax exchange; the NOTE says what.