Result · DS-EFFICIENT-SCORE-DOMINATION

Efficient scores bound profiled information

Proved herepublication: internal

Meaning

Between-cell information of the full-data efficient score dominates the profiled information for the same rule.

Scientific context

For any rule, the profiled information it retains is dominated by the between-cell information of the full-data efficient score under that same rule. Maximizing the latter over all rules with the same number of cells therefore gives a computable ceiling for the former — a number that can be subtracted from an achieved objective to see how much room is left — and the maximizer also supplies a strong initializer for the search.

The variational identity underneath this domination bound is re-attributed to Krein, and to Anderson, and to Li and Mathias, and its statistical reading is textbook semiparametrics. This is the highest-risk attribution among the results discussed here, and it is treated as a corollary of known variational and semiparametric theory rather than as a contribution.

Statement

For full-data efficient score S_hat=S_ψ-B* S_λ, every quantizer satisfies Schur_ψ(I_q) <= Var(E[S_hat|q(S)]) in Loewner order.

Assumptions

full-data nuisance block regular enough to define B*

Closest prior work

Nearest prior work

Alsing, Wandelt (2019)Nuisance Hardened Data Compression for Fast Likelihood-Free Inference

Relationships

Rests on

Enables

Raises

Systematic template morphing and nuisance scalabilityOPEN-HEP-NUISANCE-SCALING

Open next door

In the library

Carried by EfficientScoreBound. Reference (opens in a new tab).

Every object and refusal

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.

From Profiled Ds-optimality · chapter contents