= Allocation of a deterministic bias budget among baseline constraints
{title2=$\delta_j=f_jE/B_j,\quad\sum_jf_j\leq1$}
\b[Allocation of a deterministic bias budget among baseline constraints] assigns how much residual signal-response and background-leakage error each constraint may contribute. Let the allowed total bias be $E>0$, and let coefficient bounds $B_j>0$ multiply response errors $r_j$. Choose nonnegative fractions $f_j$ with $\sum_jf_j\leq1$ and set
$$
\delta_j=\frac{f_jE}{B_j}.
$$
If every computed response obeys $|r_j|\leq\delta_j$, the triangle inequality gives $|\operatorname{Bias}|\leq\sum_jB_j\delta_j\leq E$. The <response-normalized tolerance for baseline-feasibility residuals> checks those component limits. Positive tolerances require positive allocated fractions; a zero allocation demands exact cancellation in the stated model.
For an original example, take $E=12\,\mathrm{nV}$ and bounds of $2\,\mathrm{mV}$ for signal amplitude, $10\,\mathrm{mV}$ for constant background and $100\,\mathrm{V/s}$ for linear drift. Allocate fractions $1/6$, $1/3$ and $1/2$. The respective bias shares are $2$, $4$ and $6\,\mathrm{nV}$, giving tolerances
$$
\delta_s=10^{-6},\qquad\delta_0=4\times10^{-7},\qquad
\delta_1=6\times10^{-11}\,\mathrm s.
$$
Equal bias shares would instead allow $4\,\mathrm{nV}$ each, relaxing the signal-response tolerance to $2\times10^{-6}$ while tightening the drift tolerance to $4\times10^{-11}\,\mathrm s$. Both allocations guarantee the same total bound when their assumptions hold; their numerical and experimental demands differ.
An allocation should reflect attainable constraint accuracy and the cost of tightening each response. <Cost-weighted allocation of baseline-response tolerances> can formalize that tradeoff. Equal numerical tolerances across differently dimensioned conditions do not mean equal physical bias contributions. If a bound $B_j$ is uncertain, use a justified upper bound or include its uncertainty explicitly before calling the total a guarantee.
NIST's pulse-energy apparatus used before/after measurements for baseline and drift correction.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.5, printed p. 8; §7.2, Leakage Effect]{ref} The bounds above are illustrative design choices. This budget controls deterministic leakage within the assumed background family. It is separate from measurement-noise <variance>, whose <covariance> contributions must still be propagated.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
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