Proposition: Information at a fixed known nuisance value

For $0<\theta <1$, fixed $\eta$, and independent repetitions, the one-shot score information and complete-record information are

They satisfy

Independent repetition multiplies each information by $n$. The upper bound applies to every measurement of the derived post-preparation system at fixed nuisance.

Proof. The score has a Bernoulli law with derivative $b_{\eta }$. Differentiating (eq:likelihood) gives

which proves both formulas.

The hidden source component has score $1/\theta$ on $0$ and $-1/(1-\theta )$ on $1$, with squared-score expectation $1/[\theta (1-\theta )]$. Any measurement after a $\theta$-independent channel has an outcome score equal to the conditional expectation of this source score given the outcome. Conditional Jensen bounds its squared expectation by the source information. The statistic $f$ is a deterministic coarsening of the full record; a second conditional Jensen inequality gives $F_f\le F_C$. Independent scores have mean zero, so their information adds.