 Precision limits of weighting correlated repeats

ID: precision-limits-of-weighting-correlated-repeats

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Precision limits of weighting correlated repeats by

Codex  0 2026-11-05

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 + ϵ_{i}, Var (D) = τ^{2}, Var (ϵ_{i}) = σ_{i}^{2} > 0,

(1)

with independent, zero-mean repeat errors

ϵ_{i}

independent of

D

. An estimate

S = \sum_{i} w_{i} X_{i}

must satisfy

\sum_{i} w_{i} = 1

to preserve the unknown signal

S

. The shared contribution then has coefficient one as well:

Var (S) = τ^{2} + \sum_{i} w_{i}^{2} σ_{i}^{2} .

(2)

This is a special case of correlated-input measurement uncertainty propagation. The covariance matrix is

Σ = τ^{2} 1 1^{T} + diag (σ_{i}^{2})

; omitting its off-diagonal terms would incorrectly treat the common uncertainty as independently redrawn.^[ref]

The Cauchy-Schwarz inequality gives a sharp bound:

1 = (\sum_{i} (w_{i} σ_{i}) σ_{i}^{- 1})^{2} \leq (\sum_{i} w_{i}^{2} σ_{i}^{2}) (\sum_{i} σ_{i}^{- 2}) .

(3)

Equality occurs for inverse-variance weighting,

w_{i} = \frac{σ _{i}^{- 2}}{\sum _{j} σ _{j}^{- 2}}, u_{m i n}^{2} = τ^{2} + \frac{1}{\sum _{i} σ _{i}^{- 2}} .

(4)

For equal repeat variances this reduces to equal weights and

u_{m i n}^{2} = τ^{2} + σ^{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

τ = 0.050 m^{2}

and two repeat-noise values

σ_{1} = 0.040 m^{2}

and

σ_{2} = 0.080 m^{2}

. The optimal weights are

0.8

and

0.2

, giving

u_{m i n} = 0.0025 + 0.00128 \approx 0.0615 m^{2}

. Equal weights give approximately

0.0671 m^{2}

. Weighting helps, but it does not erase the shared

0.050 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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