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Open question · OPEN-D-EFFICIENCY-VS-K

How many bins meet an information target?

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Meaning

Sharp efficiency bounds as a function of bin count remain unresolved.

Scientific context

There is no sharp bound on efficiency as a function of the cell count, and no inversion rule giving a required cell count for a target efficiency. The rank ceiling (FI-RANK-CEILING) gives a minimum cell count; everything above that minimum is empirical rather than derived from a theorem.

Statement

Find distribution-dependent/distribution-free bounds on eta_D(K) and inversion rules for required bin count at target efficiency.

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Local map of Sharp D-efficiency bounds versus K1 prerequisites to the left, 0 results it enables or questions it raises to the right, 0 boundary cases below.D-efficiency relative to unbinned inferenceD-efficiency relative…Sharp D-efficiency bounds versus KSharp D-efficiency bo…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Sharp D-efficiency versus K

Study ηD(K)=sup⁡∣q∣=K(det⁡Iq/det⁡Ifull)1/d\eta_D(K)=\sup_{|q|=K}(\det I_q/\det I_{\rm full})^{1/d}. Find distribution-dependent or distribution-free bounds and inversion formulas for "how many bins for a target efficiency".

From OPEN_PROBLEMS.md