Theorem. Total-variation transport bound [ftip-00GM]
✍️source

For any score \(s:\mathcal Z\to [0,1]\) and finite laws \(\mathsf Q,\mathsf Q'\), \(|\mathbb E_{\mathsf Q}s-\mathbb E_{\mathsf Q'}s|\leq \lVert \mathsf Q-\mathsf Q'\rVert _{\rm TV}\), where \(\lVert \mathsf Q-\mathsf Q'\rVert _{\rm TV}:=\frac 12\sum _{z\in \mathcal Z} |\mathsf Q(z)-\mathsf Q'(z)|\).

Proof.

Expand the finite expectation difference and use \(|s(z)|\leq 1\). The positive and negative parts are bounded by the total variation mass, giving the displayed inequality.

Theorem. Total-variation transport bound [ftip-00GM]
✍️source

Proof.

Distribution shift and ranking reversals [ftip-00GK]

Example. Train and evaluation laws can reverse a ranking [ftip-00GL]

Theorem. Total-variation transport bound [ftip-00GM]
✍️source

Proof.

Example. A sharp two-point transport bound [ftip-00GN]

Remark. What transport does not identify [ftip-00GO]

Example. A minimal evaluation trace [ftip-00GP]

Remark. Execution controls for a matched evaluation [ftip-00GQ]

Theorem. Utility under a change of evaluation law [ftip-00IB]