= Precision limits of weighting correlated repeats
\b[Precision limits of weighting correlated repeats] describes what a <weighted mean> can achieve when repeated observations share an uncertain contribution. Suppose
$$
X_i=S+D+\epsilon_i,\qquad \operatorname{Var}(D)=\tau^2,\qquad
\operatorname{Var}(\epsilon_i)=\sigma_i^2>0,
$$
with independent, zero-mean repeat errors $\epsilon_i$ independent of $D$. An estimate $\widehat S=\sum_iw_iX_i$ must satisfy $\sum_iw_i=1$ to preserve the unknown signal $S$. The shared contribution then has coefficient one as well:
$$
\operatorname{Var}(\widehat S)=\tau^2+\sum_iw_i^2\sigma_i^2.
$$
This is a special case of correlated-input <measurement uncertainty> propagation. The <covariance matrix> is $\Sigma=\tau^2\mathbf1\mathbf1^{\mathsf T}+\operatorname{diag}(\sigma_i^2)$; omitting its off-diagonal terms would incorrectly treat the common uncertainty as independently redrawn.<Guide to the expression of uncertainty in measurement by JCGM>[§5.2.2, equation13, and §5.2.4, common instruments and reference data]{ref}
The <Cauchy-Schwarz inequality> gives a sharp bound:
$$
1=\left(\sum_i(w_i\sigma_i)\sigma_i^{-1}\right)^2
\le\left(\sum_iw_i^2\sigma_i^2\right)\left(\sum_i\sigma_i^{-2}\right).
$$
Equality occurs for <inverse-variance weighting>,
$$
w_i=\frac{\sigma_i^{-2}}{\sum_j\sigma_j^{-2}},\qquad
\boxed{u_{\min}^2=\tau^2+\frac1{\sum_i\sigma_i^{-2}}.}
$$
For equal repeat variances this reduces to equal weights and $u_{\min}^2=\tau^2+\sigma^2/n$. Allowing negative weights cannot improve this bound within the stated model.
For an original <body surface area> prediction example, take shared <standard uncertainty> $\tau=0.050\,\mathrm{m^2}$ and two repeat-noise values $\sigma_1=0.040\,\mathrm{m^2}$ and $\sigma_2=0.080\,\mathrm{m^2}$. The optimal weights are $0.8$ and $0.2$, giving $u_{\min}=\sqrt{0.0025+0.00128}\approx0.0615\,\mathrm{m^2}$. Equal weights give approximately $0.0671\,\mathrm{m^2}$. Weighting helps, but it does not erase the shared $0.050\,\mathrm{m^2}$ contribution. These values are stipulated, not clinical performance data.
For a fixed individual, $D$ can represent an unknown constant model discrepancy rather than a fluctuation between readings. No weights summing to one can distinguish that discrepancy from $S$, because both enter every observation identically. Independent reference information or a better anatomical <measurement model> is required to constrain it. If the residual errors themselves have further <covariance>, their full matrix replaces the diagonal repeat-noise model; the displayed inverse-variance weights no longer follow automatically.
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