Result · DS-EFFICIENT-SCORE-GLOBAL-UPPER

An efficient-score problem bounds profiled optimization from above

Proved herepublication: internal

Meaning

Optimizing efficient-score information over the same bin budget bounds the global profiled optimum.

Statement

Taking determinants/suprema in efficient-score domination yields a lower-dimensional D-optimal quantization problem whose optimum is an upper bound on the globally optimal in-bin Ds value.

Relationships

Rests on

Full-data efficient-score dominationDS-EFFICIENT-SCORE-DOMINATION

The full chain beneath it

Enables

Open next door

Local graph

Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

DS7. Full-data efficient-score domination

Let

\widehat S=S_\psi-B^*S_\lambda, \qquad B^*=I^{\rm full}_{\psi\lambda}(I^{\rm full}_{\lambda\lambda})^{-1}.

For every quantizer $q$ ,

\boxed{ S_\psi(I_q) \preceq \operatorname{Var}(E[\widehat S\mid q(S)]). }

Interpretation: profiling nuisance information from the bins cannot beat first projecting to the full-data efficient score and then measuring the retained between-cell information of that projection.

Taking log-determinants and suprema gives

\boxed{ \sup_q F_s(q) \le \text{best D-optimal K-bin value for the }d_\psi\text{-dimensional efficient score}. }

For the last equality to deterministic quantization of $\widehat S$ , atomlessness is required for the efficient-score law itself.

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