= Eigenmode alignment in overlap variance correction
<Eigenmode alignment in overlap variance correction> describes how shared calibration directions change the mean of a variance-corrected overlap, even when the marginal distributions of the reading and estimated variance are fixed. In the <quadratic Gaussian overlap correction> model, let $Z$ have the standard <multivariate normal distribution>, $d=a^{\mathsf T}Z$ and $\widehat v=c+Z^{\mathsf T}BZ$, with symmetric <positive semidefinite matrix> $B$ and $c\ge0$. The corrected estimate is $\widetilde\gamma=\gamma\exp(d-\widehat v/2)$.
Write $B=Q\operatorname{diag}(\lambda_k)Q^{\mathsf T}$ with <orthogonal matrix> $Q$, and let $\alpha=Q^{\mathsf T}a$. Completing the square in the Gaussian density gives
$$
\log\frac{\mathbb E[\widetilde\gamma]}\gamma
=-\frac c2-\frac12\sum_k\log(1+\lambda_k)
+\frac12\sum_k\frac{\alpha_k^2}{1+\lambda_k}.
$$
The <eigenvalues> determine the variance estimator's distribution. Its <eigenvectors> determine which of those directions also enters $d$. Rotating $B$ relative to a fixed $a$ preserves those <eigenvalues> but changes the last sum.
An original rank-one example takes $\|a\|=1$, $B=0.2uu^{\mathsf T}$ with unit vector $u$, and $c=0.8$. If $\theta$ is the angle between $a$ and $u$, direct inversion of $I+B$ gives
$$
\log\frac{\mathbb E[\widetilde\gamma]}\gamma
=0.1-\tfrac12\log(1.2)-\frac1{12}\cos^2\theta.
$$
Parallel alignment gives a mean ratio about $0.9282$; perpendicular alignment gives about $1.0089$. Both estimated variances have mean one and distribution $0.8+0.2Z_1^2$. Unbiased estimation of the logarithmic variance therefore does not itself make the corrected arithmetic overlap unbiased.
A <Rayleigh quotient> bounds every orientation when $a\ne0$:
$$
\frac{\|a\|^2}{1+\lambda_{\max}}
\le a^{\mathsf T}(I+B)^{-1}a
\le\frac{\|a\|^2}{1+\lambda_{\min}}.
$$
The endpoints occur when $a$ lies in the respective extreme <eigenspaces>. Substituting these bounds into the logarithmic mean formula gives a deterministic range over all alignments with the stated <eigenvalues>.
There is an instructive dependence trap. Symmetry gives $\operatorname{cov}(d,\widehat v)=0$, while the Gaussian fourth moments give
$$
\operatorname{cov}(d^2,\widehat v)=2a^{\mathsf T}Ba.
$$
Zero linear <covariance> between the signed residual and the variance estimate therefore does not establish that they are <independent random variables>. Squaring the residual exposes shared directions.
For an actual calibration, the joint input model must describe which records are reused. JCGM 101 separates assigning input distributions from propagating them and presents the <multivariate normal distribution> as one possible joint assignment.<Propagation of distributions using a Monte Carlo method by JCGM>[§§5.1–5.2 and 6.4.8]{ref} The examples and orientation bounds above are derived probability models, not measured heterodyne performance.
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