Zeta Sample Information Additivity

Abstract

Independent zeta observations add their Fisher information exactly.

Theorem 1.1 (Independent zeta samples have additive information).

$$\forall s:\mathbb{R},m:\mathbb{N},1<s\Rightarrow \mathrm{VarianceUnder}\left(\mathrm{ProductZetaLaw}\left(s,m\right),\mathrm{SumOfLogCoordinates}\left(m\right)\right)=\mathrm{Product}\left(m,\mathrm{VarianceUnder}\left(\mathrm{ZetaLaw}\left(s\right),\mathrm{LogObservation}\left(\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaObservation/ZetaSampleInformationAdditivity.zeta_sample_information_additive (✓ std3). ∎

Source. Repository-derived.

Commentary.

The one-sample Fisher information is represented by the variance of the logarithmic observation under the zeta law. The m-sample quantity is the variance of the sum of the m coordinate observations under the canonical product zeta measure.

Above inverse temperature one, the logarithmic observation has a finite second moment. Variance additivity for the independent product coordinates then makes the joint information exactly m times the one-sample information, including the zero-sample case.

References

• Truth anchor: D5/S3/Analytic/ZetaObservation/ZetaSampleInformationAdditivity.zeta_sample_information_additive