Proposition: Non-identifiability of omitted parameters

Let $\zeta$ denote any additional parameter that is absent from ${\rho }_{\theta }$, $\Phi$, the readout, and the declared repeated-shot model. Then the complete record law is independent of $\zeta$. Where derivatives are defined, every classical Fisher-information entry involving $\zeta$ is zero.

No inference procedure on these records distinguishes two values of $\zeta$ with different output distributions.

Proof. The complete record probability is

which contains no $\zeta$. Its $\zeta$-derivative is zero. A classical post-processing of identical input laws has identical output laws.