Correlated Systematics and Shared Inputs

Gravity tests that use different instruments are not automatically independent. They can share clock standards, Solar-System ephemerides, calibration campaigns, cosmological parameters, source populations, simulations, atomic data, or theory priors. Representing a common uncertain input as independent errors changes both combined precision and tension. The direction of the error depends on the response signs and on whether the statistic averages or differences the measurements.

Required background. Probability and Inference for Gravity supplies probability and statistical inference, covariance, nuisance marginalization, and hierarchical likelihoods.

Helpful background. Weak Metric Perturbations and Gauge Freedom supplies weak-field gravity parameters. The Tests of Gravity overview supplies the experiment map. Gravitational-Wave Astronomy supplies gravitational-wave data analysis and inference, calibration, and selection concepts.

Shared inputs generate off-diagonal covariance

Suppose experiment $j$ measures

$$y_j=f_j(\theta)+a_j\lambda+\epsilon_j,$$

where $\theta$ is a gravity parameter, $\lambda$ is a shared calibration or model input with variance $\sigma_\lambda^2$, and the $\epsilon_j$ are independent errors. Then

$$\operatorname{Cov}(y_i,y_j) =a_i a_j\sigma_\lambda^2+\delta_{ij}\sigma_{\epsilon_i}^2.$$

Adding $a_j^2\sigma_\lambda^2$ separately to diagonal error bars loses the off-diagonal term. If $a_i$ and $a_j$ have the same sign, a common shift cannot be averaged down by adding experiments; if their signs differ, a difference statistic can amplify it.

The more transparent formulation retains the shared latent variable,

$$p(\{d_j\}\mid\theta) =\int d\lambda\,\pi(\lambda) \prod_j\left[\int d\eta_j\, p(d_j\mid\theta,\lambda,\eta_j)\pi(\eta_j\mid\lambda)\right].$$

Here $\eta_j$ are experiment-specific nuisance parameters. This expression prevents the prior on $\lambda$ from being counted once per dataset and allows data to update the common input coherently.

Where correlations enter gravity tests

Planetary ranging, spacecraft radio science, pulsar timing, and some clock analyses can share a Solar-System ephemeris and time standard. Lensing, clustering, and standard-siren comparisons can share redshift catalogs, cosmological distance assumptions, photometric calibration, and nonlinear simulations. Gravitational-wave events in one observing epoch share calibration models and population selections. Black-hole imaging, X-ray spectroscopy, and stellar orbits may share distance and mass information even though their instruments differ.

Theory mapping creates another correlation. Reusing a published posterior as a likelihood carries its original prior unless the prior density is divided out in the same coordinates and support. Posterior samples from two analyses can also contain duplicate events, catalogs, or simulated training data. Multiplying them without provenance is not hierarchical inference; it is repeated use of information.

Selection is shared when the same survey or catalog supplies targets to several tests. If a gravity-dependent observable helped select the sample, later treating selection as independent of the gravity parameter biases the population likelihood. The release identifier, mask, quality cuts, completeness model, and object cross-match are required to diagnose the overlap Mandel, Farr, and Gair 2019, §§2–4.

Results and limitations through 2026-08-19

At the 2026-08-19 cutoff, individual mature analyses often propagate important internal covariance, but no universal public covariance product spans every gravity regime. Cross-regime synthesis therefore remains limited by provenance and shared-input modeling as much as by statistical precision.

Evidence or result	Status at the cutoff	Limitation or uncertainty boundary
Major experiment-level releases document calibration and internal covariance	Strong basis for within-release inference	Cross-release and cross-field covariance is rarely distributed in a common form
Hierarchical methods correctly represent shared population and selection parameters	Established statistical construction	Published marginalized posteriors may not retain the likelihood or shared nuisance dimensions
Independent instruments provide complementary response directions	Real gain when inputs and samples are disjoint	Common standards, ephemerides, catalogs, simulations, and priors prevent naive independence
Apparent cross-regime discrepancies can be quantified with concordance diagnostics	Useful evidence-localization tools exist	Different diagnostics answer different questions and depend on covariance completeness

The optimistic position is that heterogeneous instruments supply nearly independent confirmation, so their combined precision can grow rapidly. The conservative competing position is that the dominant uncertainty is often a shared forward-model input, making the nominal gain illusory. Negative evidence for naive independence is a result that moves coherently when one ephemeris, distance scale, calibration basis, or simulation suite is changed.

Raveri and Hu distinguish goodness-of-fit loss, evidence-ratio calibration, and parameter-shift estimators; agreement or disagreement among them should be interpreted through the quantity each tests Raveri and Hu 2019, §II.C–E. None can reconstruct a covariance source that the released products omitted.

A provenance-first combination workflow

For every input, record dataset or catalog release, calibration and software version, retrieval date, file hash, selection mask, and prior density. Build an overlap graph whose nodes are analyses and whose edges name shared events, standards, nuisance parameters, simulations, or priors. “Possibly correlated” is not a covariance model, but the graph identifies which products cannot be multiplied safely.

Then perturb each common input jointly. An independent check changes a shared ephemeris, calibration eigenmode, cosmological parameter, or simulation hyperparameter once and recomputes all affected observables. The resulting coherent shifts estimate response vectors $a_j$ and reveal sign errors or hidden duplication. A block bootstrap over shared objects can expose catalog overlap that analytic covariance misses.

Reproduction requires a machine-readable input inventory and one likelihood evaluation at benchmark parameter values. Compare a model with explicit shared latent variables to a deliberately independent model. If the latter tightens the gravity interval far more than genuine complementarity predicts, the difference is a diagnostic of duplicated information.

An adversarial test assigns the largest plausible common covariance consistent with calibration data and asks whether the claimed combined exclusion or tension survives. This is not permission to choose an arbitrary correlation after seeing the answer; the covariance family and acceptance criterion must be predeclared.

The unresolved issue is practical and model-dependent: many legacy products do not expose the likelihood factors or nuisance samples needed for exact recombination. A precise global constraint assembled from such products can be less reliable than a wider result using fewer inputs whose assumptions can be checked.

What could change the conclusion. The interpretation could change when a major release supplies a new cross-analysis covariance product, when a duplicated catalog or prior is discovered, or when the joint perturbation test changes a published cross-regime conclusion beyond its stated uncertainty.

Check your understanding

Two measurements have independent noise variance $1$ and share an additive calibration offset with variance $4$. What covariance matrix should be used, and why does averaging the measurements not reduce the calibration part?

Solution

With unit response to the common offset, the covariance is $C=I+4\boldsymbol 1\boldsymbol 1^T$, so its diagonal entries are $5$ and its off-diagonal entries are $4$. The average contains the same offset once: its independent-noise variance becomes $1/2$, but its calibration variance remains $4$. Therefore treating the two total variances as independent would incorrectly give $5/2$ and claim that the shared standard averaged down. Introducing one offset parameter with its prior gives the same covariance after marginalization and makes the physical reason explicit.

References

• 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.
• Raveri, M., and Hu, W. (2019), “Concordance and discordance in cosmology,” Physical Review D 99, 043506. Raveri and Hu 2019, §II.C–E.

Further reading

• Particle Data Group (2024), “Statistics,” in Review of Particle Physics, Physical Review D 110, 030001. Particle Data Group 2024, §§40.1–40.4, PDF.