Nuisance Parameters and Systematic Error

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A nuisance parameter is an uncertain quantity needed to predict the data but not itself the target of the gravity test. A systematic error is not merely “a nuisance with a name”: it is a bias or loss of coverage caused by a missing, misspecified, or incorrectly propagated part of the measurement model. The distinction matters because marginalization can propagate uncertainty in a represented effect, but it cannot repair an effect absent from the likelihood.

Required background. The Einstein Equation supplies the Einstein equations underlying the reference prediction. Probability and Inference for Gravity supplies likelihoods, priors, covariance, sampling distributions, and marginalization.

Helpful background. Weak Metric Perturbations and Gauge Freedom supplies weak-field signal parameters used in examples. The Tests of Gravity overview connects these statistical controls to the experimental chapters.

Nuisance parameters inside the likelihood

Write the measured data vector as

\boldsymbol d =\boldsymbol s(\theta,\eta) +\boldsymbol b(\eta) +\boldsymbol n,

where $\theta$ is the gravity parameter of interest, $\eta$ collects source, calibration, background, and population quantities, and $\boldsymbol n$ is stochastic noise. For a Gaussian approximation,

-2\ln p(\boldsymbol d\mid\theta,\eta) =\bigl[\boldsymbol d-\boldsymbol\mu(\theta,\eta)\bigr]^T C^{-1}(\eta) \bigl[\boldsymbol d-\boldsymbol\mu(\theta,\eta)\bigr] +\ln\det C(\eta)+\text{constant}.

Both the mean $\boldsymbol\mu$ and covariance $C$ can depend on nuisance parameters. Dropping $\ln\det C$ while allowing $C$ to vary does not define the same normalized likelihood. Treating a shared calibration scale as an independent error bar for every datum double-counts information; it should instead be one common parameter, or a correlated process with a documented covariance kernel.

Profiling and marginalization answer related but distinct questions. A profile likelihood maximizes over $\eta$ for each $\theta$ . A Bayesian posterior integrates over $\eta$ with a prior measure. Agreement in a regular, informative limit is reassuring, but disagreement near boundaries or along weakly identified directions is diagnostic rather than embarrassing.

Calibration, selection, and model discrepancy

Calibration maps an instrument output to a physical quantity. Its transfer function may have amplitude, phase, time, wavelength, angular, or reference-frame uncertainty. The calibration version must be pinned to the dataset or data release; a later recalibration can move both the central value and covariance. Calibration data should enter jointly when possible, rather than being compressed into a single percentage that erases correlations.

Selection changes the population represented by the sample. If objects with latent parameters $x$ are detected with probability $P_{\rm det}(x)$ , a population likelihood contains the selection normalization

\alpha(\Lambda)=\int P_{\rm det}(x)\,p(x\mid\Lambda)\,dx,

where $\Lambda$ denotes population and possibly gravity parameters. Omitting the $\Lambda$ dependence can bias a gravity parameter whenever the proposed deviation changes detectability. Quality cuts, masks, event thresholds, and rejected epochs are selections too; completeness must be defined for the population to which the conclusion is generalized Mandel, Farr, and Gair 2019, §§2–4.

Model discrepancy describes the difference between an approximate forward model and the physical signal even at the best parameter values. Waveform truncation, unmodeled plasma, imperfect ephemerides, baryonic feedback, and finite numerical resolution are examples. Inflating diagonal noise can hide residual structure without representing it. A better treatment introduces a physically constrained discrepancy model, compares independent approximants, or excludes the region where no controlled prediction exists.

Diagnose bias before quoting precision

Failure mode	Observable symptom	Required control	What not to claim
Shared calibration omitted	Coherent shift across channels	Joint calibration parameter and covariance	Independent confirmation from correlated channels
Selection ignored	Parameter drift with threshold	Injection-based efficiency and population normalization	An unbiased population constraint
Approximation error	Residual follows frequency or scale	Multiple controlled models and domain cut	New gravity from model-shaped residuals
Prior-boundary effect	Limit tracks prior endpoint	Reparameterization and coverage/sensitivity study	Data-dominated exclusion

Synthetic-data tests are indispensable. An injection study passes known null and alternative signals through the complete selection and inference pipeline. Simulation-based calibration checks posterior rank uniformity over draws from the declared generative model. Frequentist coverage checks whether nominal intervals contain the injected value at the advertised rate. An Asimov dataset—defined by expected data rather than a random realization—can forecast median sensitivity, but it does not replace fluctuations or misspecification tests Cowan et al. 2011, §3.2, Eq. (24), Eqs. (24)–(30), and §4.1.

Blind analysis helps when flexible cuts or calibration choices can be tuned toward a desired answer. The blinding rule, unblinding criteria, pipeline version, and any post-unblinding changes should be recorded. Reproducibility requires the release identifier, retrieved files or hashes, software environment, configuration, random seeds, and a machine-readable table linking every selection to its effect on the result.

Interpreting the results and their limits

This methodological account uses an evidence cutoff of 2026-08-19. A result cited elsewhere in the volume must identify the dataset/catalog release, calibration and pipeline version, selections, statistical model, and known systematic limitations current at that date. A superseding release is a reason to review the affected numerical conclusion, not to silently mix versions.

A robust gravity statement should report how the result changes under justified nuisance alternatives, not only the smallest error bar from a preferred model. Stability is not obtained by trying many analyses and selecting the most favorable one. The analysis family and acceptance criterion should be fixed, and unexplained spread should enter the uncertainty or narrow the claim.

Check your understanding

Two instruments share the same time standard, whose offset has uncertainty $\sigma_t$ . Why is adding $\sigma_t^2$ independently to both diagonal variances wrong, and how should the likelihood represent the uncertainty?

Solution

The same clock offset moves both measurements coherently, so the induced errors are correlated. For equal response to the offset, independent noise variance $\sigma^2$ , and $y_i=\theta+\delta t+\epsilon_i$ , the correct covariance and its two useful combinations are

\begin{aligned} C&=\sigma^2 I+\sigma_t^2\boldsymbol 1\boldsymbol 1^T,\\ \operatorname{Var}\!\left(\frac{y_1+y_2}{2}\right) &=\frac{\sigma^2}{2}+\sigma_t^2,\\ \operatorname{Var}(y_1-y_2)&=2\sigma^2. \end{aligned}

The common offset survives in the average and cancels from the difference. Replacing it by independent diagonal errors instead gives $(\sigma^2+\sigma_t^2)/2$ for the average and $2(\sigma^2+\sigma_t^2)$ for the difference: it makes the average too precise and the difference too uncertain. Introduce one offset parameter with its calibration prior, or use the rank-one covariance $\sigma_t^2\boldsymbol a\boldsymbol a^T$ for general responses $\boldsymbol a$ . Integrating out that one Gaussian offset gives the covariance form and supplies an independent check.

References

Cowan, G., Cranmer, K., Gross, E., and Vitells, O. (2011), “Asymptotic formulae for likelihood-based tests of new physics,” European Physical Journal C 71, 1554. Cowan et al. 2011, §3.2 and §4.1.
Mandel, I., Farr, W. M., and Gair, J. R. (2019), “Extracting distribution parameters from multiple uncertain observations with selection biases,” Monthly Notices of the Royal Astronomical Society 486, 1086–1093. Mandel, Farr, and Gair 2019, §§2–4.