Source Clarifications: A No Free Lunch Theorem for Human-AI Collaboration

Loss, correctness, and calibration

• Opening “accuracy” $\mathbb E[|\widehat Y(X)-Y|]$ → zero-one loss. Expected correctness is its complement $1-\mathbb E[|\widehat Y(X)-Y|]$.

Mixtures and finite partitions

• Proposition 6 component-$m$ sampling weight $\lambda_\ell$ → $\lambda_m$. The component’s joint law and accuracy equations use the same simplex weight.
• Undefined label-frequency quotient on a null partition cell → zero in the finite predictor-realization construction used by Lemma 8 and Proposition 9. This fills an otherwise undefined branch.

Proposition 9’s two settings

• First setting’s “small positive” $\epsilon$ → $0<\epsilon<1/2$ to keep the constructed prediction profiles interior.

• Second setting’s free-index denominator → $2+\sum_j[p_j/(1-p_j)+(1-q_j)/q_j]$, where $p,q$ are the compared interior prediction profiles. Both odds families must be summed to define the normalized finite probability law.

• Proposition 9 mixture “sufficiently close to one” → weights $7/8$ and $1/8$, which satisfy its strict comparison.