Hierarchical Multi-Experiment Likelihoods

A hierarchical likelihood combines heterogeneous gravity tests by separating common theory parameters, population variation, shared inputs, and experiment-specific nuisance quantities. It is not synonymous with multiplying published posteriors. The hierarchy must state what is universal, what varies from source to source, how each theory prediction enters each experiment, and how selection changes the observed sample.

Required background. Probability and Inference for Gravity supplies probability and statistical inference, latent variables, hierarchical likelihoods, priors, and posterior sampling.

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

Common parameters and local latent variables

Let $\alpha$ denote parameters of one gravitational theory and $\lambda$ shared calibration, cosmology, or boundary inputs. Experiment $j$ has local physical variables $x_j$ and nuisance parameters $\eta_j$. Suppose the local latent variables are independent across experiments conditional on $(\alpha,\lambda)$, and the data are independent conditional on those shared and local variables. A coherent likelihood is then

$$p(\{d_j\}\mid\alpha) =\int d\lambda\,\pi(\lambda\mid\alpha) \prod_j\int dx_j\,d\eta_j\, p(d_j\mid F_j(\alpha,\lambda,x_j),\eta_j) p(x_j,\eta_j\mid\alpha,\lambda).$$

The maps $F_j$ can differ across laboratory, Solar-System, pulsar, wave, galactic, and cosmological regimes. Universality resides in $\alpha$ and any theory-prescribed relations, not in forcing all fitted phenomenological coefficients to have the same value.

Population variation can be explicit. A gravitational-wave deformation $\delta_i$ might be modeled as draws from a distribution with mean $\mu$ and width $\sigma$. The special case $\sigma=0$ asserts that every source has exactly one common value. That assertion is much stronger than “combine the events” and can produce an apparently tight constraint when the true mapping depends on mass, spin, redshift, or environment Isi, Chatziioannou, and Farr 2019, main hierarchical construction.

Selection belongs in the normalization

For an observed catalog with latent parameters $x_i$ drawn from a normalized population $p(x\mid\Lambda)$, let $R$ be the expected number of sources before selection during the stated observation, not a dimensionful rate density. Assume independent sources drawn from a Poisson process, and detection determined by the recorded data. Then

$$\mathcal L(\{d_i\},N_{\rm obs}\mid R,\Lambda) \propto R^{N_{\rm obs}}e^{-R\,\xi(\Lambda)} \prod_{i=1}^{N_{\rm obs}} \int p(d_i\mid x_i)p(x_i\mid\Lambda)\,dx_i,$$

Here $\xi(\Lambda)=\int P_{\rm det}(x)p(x\mid\Lambda)\,dx$ is a dimensionless efficiency and $R\xi$ is the expected detected count. The factors $p(d_i\mid x_i)$ are the unconditioned data likelihoods, evaluated at data that passed the selection; do not multiply them by $P_{\rm det}(x_i)$ again. A physical rate per comoving volume and source time requires the corresponding spacetime measure in the intensity.

Conditioning on the observed count instead gives

$$\mathcal L(\{d_i\}\mid N_{\rm obs},\Lambda) \propto\xi(\Lambda)^{-N_{\rm obs}} \prod_i\int p(d_i\mid x_i)p(x_i\mid\Lambda)\,dx_i.$$

One way to recover this shape dependence is to integrate the rate-explicit expression with $\pi(R)\propto1/R$ for $N_{\rm obs}\gt0$: the integral is $\Gamma(N_{\rm obs})/\xi^{N_{\rm obs}}$. A flat prior on $R\ge0$ gives $\Gamma(N_{\rm obs}+1)/\xi^{N_{\rm obs}+1}$ instead. These improper-prior calculations illustrate the dependence on the rate convention; model evidence requires a proper, specified rate prior. Conditioning on a count and retaining its Poisson information are different scientific choices.

If gravity changes waveform duration, luminosity distance, source evolution, lensing, or the catalog observable, it can change $\xi$. Holding selection fixed to GR while inferring a deformation can bias both the population and gravity parameters. Injection campaigns must use the same detection and quality cuts as the released sample Mandel, Farr, and Gair 2019, §§2–4.

Reusing posterior samples correctly

Suppose event $i$ was analyzed with interim prior $\pi_i(x)$ and produced posterior samples $x_i^{(k)}$. Its population integral can be estimated as

$$\int p(d_i\mid x)p(x\mid\Lambda)\,dx \simeq \frac{Z_i}{n_i} \sum_{k=1}^{n_i} \frac{p(x_i^{(k)}\mid\Lambda)}{\pi_i(x_i^{(k)})},$$

where $Z_i$ is independent of $\Lambda$ for a fixed event analysis. Multiplying posterior densities without dividing by interim priors counts those priors repeatedly. Importance weights also fail when the new population occupies support poorly sampled by the interim posterior; effective sample size and tail coverage are required diagnostics.

This identity assumes that the new model changes the population density while retaining the original event likelihood. If changing gravity also changes the predicted strain or response at fixed $x$, density reweighting alone is insufficient: reevaluate that likelihood, or include its likelihood ratio when its support is available. A delta-function common-value population likewise requires a dedicated common-parameter evaluation or controlled integration; substituting a delta function into a finite sample-weight sum is not a usable estimator.

Shared events, host catalogs, calibration parameters, and simulations should appear once. Dataset release, calibration version, event and quality selections, software commit, sampler settings, interim priors, and file hashes are necessary reproduction metadata. A machine-readable likelihood or samples with complete weights are substantially more useful than a plotted one-dimensional interval.

Results and limitations through 2026-08-19

At the 2026-08-19 cutoff, hierarchical methods are standard tools for event populations and are used in current gravitational-wave tests, but cross-experiment synthesis remains dependent on access to likelihood-level products and defensible universality assumptions. The submitted GWTC-5 tests analysis uses both common-value and hierarchical combinations in named tests; those alternatives should not be conflated LVK 2026, P2500781-v17, §§2–7.

Evidence or result	Status at the cutoff	Limitation or uncertainty boundary
Selection-normalized hierarchical likelihoods	Established and analytically well defined	Rate convention, efficiency estimation, and population support must match the test
Posterior-sample recycling	Efficient when interim priors and support are known	Reweighting fails with missing tails, undocumented priors, or low effective sample size
Population gravity tests	Can distinguish a common shift from event-to-event scatter	Hyperprior, waveform systematics, event selection, and theory mapping affect the conclusion
Cross-regime theory combination	Conceptually coherent through experiment-specific maps	Public products often omit shared nuisance dimensions and exact likelihoods

The strongest optimistic position is that many null events shrink a common deformation rapidly. The competing conservative position is that an unmodeled population width or source-dependent theory map makes the common-value product overconfident. Negative evidence for the strict universal model appears when posterior-predictive event scatter exceeds the hierarchy, when a few high-weight events control the result, or when leave-one-out estimates move substantially.

Adversarial and convergence checks

An independent check fits at least three models: one exactly common coefficient, one population with inferred width, and one theory-predicted dependence on source or environment. Simulated catalogs drawn from each should recover calibrated hyperparameters after the full selection pipeline. Leave-one-event and leave-one-experiment-out analyses expose outliers and hidden duplication.

Numerical convergence requires effective sample size for every reweighted event, stable normalization under larger injection sets, multiple sampler seeds, and a reproduced likelihood value from an independent implementation. An adversarial test expands the population family or introduces a shared calibration mode fixed before inspecting the result. A gravity claim survives only if its sign and scale remain identifiable rather than being absorbed by those plausible alternatives.

The unresolved issue is model-dependent: there is no universally correct hierarchy for “deviations from GR.” A complete theory may predict exact correlations; a bottom-up search should expose rather than conceal its population assumptions.

What could change the conclusion. The interpretation could change after a major likelihood or selection release, a discovery of undocumented interim priors or duplicated events, or a validated hierarchy that changes a consequential combined gravity result under the predeclared simulation and leave-one-out checks.

Check your understanding

Why does multiplying ten event posteriors for a common coefficient count a uniform interim prior ten times, and when would that mistake still affect the result?

Solution

Each posterior is proportional to its event likelihood times the interim prior. Their product is therefore proportional to the product of likelihoods times the prior raised to the tenth power, whereas a common-parameter analysis should apply the chosen population prior once. For the same exactly uniform prior on a fixed support $S$, repeated multiplication changes only a constant: $\mathbf1_S^{10}=\mathbf1_S$, so it does not distort the normalized shape even at that support’s boundary. Problems arise for nonconstant priors, different supports or coordinates, or a desired prior extending beyond the available samples. Prior normalization also matters for model evidence. Therefore recover likelihood factors using the documented interim priors and check their support before applying the common hierarchy.

References

• Isi, M., Chatziioannou, K., and Farr, W. M. (2019), “Hierarchical test of general relativity with gravitational waves,” Physical Review Letters 123, 121101. Isi, Chatziioannou, and Farr 2019, main construction.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026), “GWTC-5.0: Tests of General Relativity,” submitted manuscript, arXiv:2607.19293, LIGO DCC P2500781-v17. LVK 2026, Table 1 and §§2–7.
• 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.