ScoreQuant

Result · FI-RANK-CEILING

A bin budget limits information rank

Derived from the literaturepublication: derived from published

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Meaning

A regular score model with K categories has retained-information rank at most K minus one.

Scientific context

With K categories and a d-dimensional parameter, the binned information has rank at most min(d, K-1), so a nonsingular answer needs at least d + 1 categories. Asking three bins to measure four parameters is not a hard optimization problem to solve poorly; it is an impossible one, and the library refuses such a configuration rather than returning a degenerate answer.

Statement

rank(I_q)<=min(d,K-1); positive-definite full-D information requires K>=d+1.

Closest prior work

Relationships

Rests on

Enables

In the library

Carried by InformationReport.effective_rank. Reference.

Enforced by the refusal Counterexample fixtureCE-DS-MARGINS-RANK-VACUITY-001: the library declines the operation and names the fixture.

Every object and refusal

Local graph
Local map of Rank ceiling for K hard categories1 prerequisites to the left, 4 results it enables or questions it raises to the right, 0 boundary cases below.Quantized Fisher-information identityQuantized Fisher-info…Full in-bin Ds profiling is singular when K<=dFull in-bin Ds profil…Projected efficient-score problem only requires K>=d_ψ+1Projected efficient-s…Frozen-rule vector retention: plug-in CLT for the geometric-mean D-retention with a consistent influence-function varianceFrozen-rule vector re…Singular binned information: the plug-in geometric-mean retention is biased upward at rate n^{-(d-r)/d}Singular binned infor…Rank ceiling for K hard categoriesRank ceiling for K ha…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

U4. Rank ceiling

Since ∑bmb=0\sum_bm_b=0,

rank⁡(Iq)≤min⁡(d,K−1).\operatorname{rank}(I_q)\le\min(d,K-1).

Therefore K≥d+1K\ge d+1 is necessary for nonsingular full-D information.

From Universal information structure · chapter contents