= Lognormal bias of a heterodyne overlap estimate
{title2=$\mathbb E\widehat\gamma=\gamma e^{v_d/2}$}
\b[Lognormal bias of a heterodyne overlap estimate] is the upward mean shift that remains when a zero-mean Gaussian logarithmic error is exponentiated. It does not undo <common-gain cancellation in heterodyne overlap uncertainty>: a perfectly shared gain cancels in every reading, while residual mismatched gains can still fluctuate.
Write the normalized positive estimate as $\widehat\gamma=\gamma e^d$, with fixed true overlap magnitude $\gamma$ and $d\sim N(\mu_d,v_d)$. The <moment-generating function> of the <normal distribution> gives
$$
\mathbb E[\widehat\gamma]=\gamma e^{\mu_d+v_d/2},\qquad
\operatorname{median}(\widehat\gamma)=\gamma e^{\mu_d},
$$
$$
\operatorname{Var}(\widehat\gamma)=\gamma^2e^{2\mu_d+v_d}(e^{v_d}-1).
$$
These are exact <lognormal distribution> identities under the stated error model. When $\mu_d=0$, the logarithmic estimate is centered correctly and the median is $\gamma$, but the arithmetic mean exceeds it by relative amount $e^{v_d/2}-1$.
For an original deliberately visible example, $\gamma=0.8$ and logarithmic standard deviation 0.2 give $v_d=0.04$ and mean about 0.81616: a 2.02% mean shift. The values are chosen statistical parameters, not uncertainties reported for a receiver. An estimate can also exceed the physical <consistency bound for separately measured heterodyne arm powers>; the noisy readout is not itself a physical overlap above one.
If $\mu_d$ and $v_d$ are known, multiplying the estimate by $e^{-\mu_d-v_d/2}$ corrects its mean in this model. <Uncertain variance in heterodyne lognormal bias correction> matters when those parameters are estimated: plugging in a noisy variance, especially from related data, is not automatically an exact correction.
Averaging logarithms has a different interpretation from averaging overlaps. For $n$ independent zero-mean log-errors of variance $v_d$, the geometric mean is $\gamma e^{\overline d}$, whose arithmetic expectation is $\gamma e^{v_d/(2n)}$. A gain component shared across the entire run does not acquire the independent-shot $1/n$ variance reduction.
The physical calibration paths determine the logarithmic covariance. NIST's heterodyne apparatus controlled the optical modes while separately correcting electrical loss and impedance mismatch.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp. 1–2, Heterodyne measurements and Eqs. 1–4]{ref} A first-order uncertainty calculation alone is not this exact distributional model; nonlinear effects and correlated inputs must be assessed explicitly.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1.2 and 5.2]{ref} State the assumed distribution and the intended summary rather than treating median, geometric mean and arithmetic mean as interchangeable.
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