Result · DS-FULL-PROFILE-K-LE-D-SINGULAR

Too few cells make full profiling singular

Proved herepublication: internal

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Meaning

The rank ceiling obstructs full in-bin profiling when the cell budget is too small for the full score dimension.

Statement

Because rank(I_q)<=K-1, ordinary full-block in-bin Ds profiling is not identifiable when K<=d.

Relationships

Rests on

The full chain beneath it

Quantized Fisher-information identityFI-QUANT-IDENTITYdepth 2

Enables

Margins dichotomy at global finite Ds optima (conditionally centered laws, scalar nuisance)OPEN-DS-MARGINS-AT-OPTIMA

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Local graph

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

Proof

DS9. The $K\le d$ feasibility split

For the full in-bin profiled formulation, $I_q$ is singular when $K\le d$ , so ordinary full-block $D_s$ profiling is not identifiable.

The projected efficient-score problem only needs enough categories to identify the $d_\psi$ -dimensional efficient score, i.e. structurally $K\ge d_\psi+1$ , if nuisance information is supplied externally.

These are different statistical problems and must be exposed separately.

From Profiled Ds-optimality · chapter contents