= Uncertain variance in heterodyne lognormal bias correction
\b[Uncertain variance in heterodyne lognormal bias correction] matters when a measured overlap is corrected using an estimated, rather than known, logarithmic variance. The known-variance correction is a nonlinear function of that estimate, so replacing the variance by an unbiased estimate does not automatically preserve unbiasedness of the corrected overlap.
Suppose $\widehat\gamma=\gamma e^d$, with $d\sim N(0,v)$, and use $\widetilde\gamma=\widehat\gamma e^{-\widehat v/2}$. If the variance estimate is independent of the overlap readout,
$$
\mathbb E[\widetilde\gamma]=\gamma e^{v/2}\mathbb E[e^{-\widehat v/2}].
$$
Even if $\mathbb E[\widehat v]=v$, <Jensen's inequality> gives $\mathbb E[e^{-\widehat v/2}]\ge e^{-v/2}$. Consequently the mean corrected estimate is at least $\gamma$ under these independence and unbiased-variance assumptions. Equality requires a nonrandom variance estimate in the ordinary finite-variance case.
An original exact example uses a variance estimate from an independent normal calibration sample, with $\nu\widehat v/v$ following a <chi-squared distribution> with $\nu$ degrees of freedom. Its Laplace transform gives
$$
\frac{\mathbb E[\widetilde\gamma]}{\gamma}
=e^{v/2}\left(1+\frac v\nu\right)^{-\nu/2}.
$$
The logarithm of this factor is $v/2-(\nu/2)\log(1+v/\nu)$, which is positive for $v>0$. With stipulated $v=1$ and $\nu=2$, the factor is $e^{1/2}/1.5\approx1.09915$: about a 9.91% remaining mean shift. This intentionally broad statistical example is not a typical receiver specification.
For fixed $v$ and large $\nu$, the logarithm is $v^2/(4\nu)+O(\nu^{-2})$. More independent calibration information reduces this effect. Merely taking more overlap readings while retaining the same uncertain variance estimate does not remove its influence.
<Dependent variance correction in heterodyne normalization> requires the joint quantity $\mathbb E[e^{d-\widehat v/2}]$ instead of the product of expectations above. A variance inferred from the same readout cannot simply be declared independent. Propagate the joint calibration and readout model, with a stated target summary. JCGM 101 treats input distributions from prior uncertainty calculations and numerical propagation explicitly.<Propagation of distributions using a Monte Carlo method by JCGM>[§§6.5 and 7]{ref}
The optical <gain-matching condition for heterodyne normalization> still concerns physical calibration paths. This calculation addresses the statistical correction of residual log-errors; it does not imply that an imperfect RF or DC calibration can be repaired solely by subtracting a fitted variance.
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