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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 $\hat{\gamma }=\gamma e^d$, with $d\sim N(0,v)$, and use $\tilde{\gamma }=\hat{\gamma }{e}^{-\hat{v}/2}$. If the variance estimate is independent of the overlap readout,Even if $\mathbb{E}[\hat{v}]=v$, Jensen's inequality gives $\mathbb{E}[e^{-\hat{v}/2}]\ge e^{-v/2}$. 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 $\nu \hat{v}/v$ following a chi-squared distribution with $\nu$ degrees of freedom. Its Laplace transform givesThe 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 $e^{1/2}/1.5\approx 1.09915$: 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 $\mathbb{E}[e^{d-\hat{v}/2}]$ 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.^[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.