Result · PROXY-TRUE-RETAINED-FI

Proxy-built labels retain the true score's information, not the proxy's

Derived from the literaturepublication: derived from published

Meaning

The true score, conditioned on proxy-derived labels, determines achieved Fisher information; proxy moments measure a different quantity.

Scientific context

Every theorem in this area is a statement about a table of numbers; it is silent about where those numbers came from. When the score is estimated — from a classifier, a learned ratio, a simulator — the information actually retained is governed by the true score conditioned on the estimated rule, not by the estimated score conditioned on itself. The two coincide only when the proxy is exact in the relevant sense; this is why estimated-score provenance never claims exact Fisher semantics.

Statement

If q is built from estimated score s_hat, the actual retained information is Var(E[s|q(s_hat)]), not Var(E[s_hat|q(s_hat)]) unless the proxy is exact in the relevant sense.

Relationships

Rests on

Quantized Fisher-information identityFI-QUANT-IDENTITY

Enables

Raises

Open next door

In the library

Carried by RatioProvenance, ScoreProvenance. 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

O4. True retained FI under an estimated score

If the quantizer uses $\hat s$ , the actual retained Fisher information is

\boxed{ \operatorname{Var}(E[s\mid q(\hat s)]), }

not $\operatorname{Var}(E[\hat s\mid q(\hat s)])$ unless $\hat s=s$ in the relevant sense.

From Score, ratio and classifier access · chapter contents