Result · FI-LOSSLESS-IFF-SCORE-MEASURABLE

When is a hard label locally lossless?

Derived from the literaturepublication: derived from published

Meaning

Fisher information is preserved exactly when the score is measurable from the label under the stated model assumptions.

I_q=I_full iff S=h(Z) almost surely, equivalently Cov(S|Z)=0 almost surely.

Relationships

The full chain beneath it

Local graph

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

U3. Local Fisher-losslessness criterion

\boxed{ I_q=I_{\rm full} \iff S=h(Z)\quad\text{a.s.} }

Equivalently $\operatorname{Cov}(S\mid Z)=0$ a.s. Generic smooth score laws cannot be exactly lossless for finite $K$ .