Waveform Marginalization and Model Uncertainty

A posterior can be numerically precise while remaining physically inaccurate if every waveform in the likelihood shares a consequential modeling error. Waveform uncertainty must therefore enter as a model choice, a discrepancy distribution, or a validated bias bound—not as an informal enlargement applied after sampling. No single prescription is universally correct because available waveform families are correlated approximations rather than independent draws from a known distribution.

Helpful background. The measured combinations are organized in Binary Parameters and Observable Combinations, and Posterior Predictive and Residual Checks explains how model failure can appear in residual data.

Waveform error inside the likelihood

Write the true signal as

$$h_{\mathrm{true}}(\theta) =h_A(\theta)+\delta h_A(\theta),$$

where $h_A$ is an available approximant and $\delta h_A$ is its discrepancy on the analyzed detector band. If the likelihood sets $\delta h_A=0$, a component of the discrepancy parallel to the waveform tangent $\partial_i h_A$ can imitate a parameter shift. In the high-signal-to-noise, small-error limit,

$$\Delta\theta^i_{\mathrm{sys}} \simeq (\Gamma^{-1})^{ij} \sum_I (\delta h_{A,I}\mid\partial_j h_{A,I})_I.$$

This expression is diagnostic, not a full posterior correction. It assumes a locally linear waveform manifold, a well-conditioned Fisher matrix, the chosen noise inner product, and an actual estimate of $\delta h_A$. The derivation and its domain are given in Cutler and Vallisneri 2007, §§ II–III.

Mismatch alone is insufficient. The noise-weighted norm $(\delta h\mid\delta h)$ measures the size of a fixed waveform discrepancy, while its direction determines the leading consequence. In the local linear approximation, write

$$\delta h=\delta h_\parallel+\delta h_\perp, \qquad \delta h_\parallel=\partial_i h\,\Delta\theta^i_{\rm sys}.$$

The tangent component $\delta h_\parallel$ is absorbed by shifting parameters; the orthogonal component satisfies $(\delta h_\perp\mid\partial_i h)=0$ and remains in the best-fit residual. Their squared norms add. A small residual can therefore coexist with appreciable parameter bias. This decomposition includes every fitted parameter, including time, phase, and amplitude when those are free; a mismatch already maximized over some of them has removed the corresponding components.

Four ways to propagate model uncertainty

Discrete model averaging

If $A$ labels waveform families, a model-averaged posterior is

$$p(\theta\mid d) =\sum_A p(\theta\mid d,A)\,p(A\mid d),$$

with

$$p(A\mid d) =\frac{Z_A\pi(A)}{\sum_B Z_B\pi(B)}.$$

This propagates disagreement among the included models after mapping their parameters to the same physical quantities and measure. It does not cover shared omissions, and the evidence weights can be dominated by prior volume or numerical evidence error.

The evidence and prior-measure dependence entering those weights follows the general gravitational-wave Bayesian construction Thrane and Talbot 2019, §§ 2–4.

Hypermodel sampling

A sampler may treat $A$ as a discrete parameter and move between waveform families. This exposes correlations between physical parameters and model identity, but it inherits the same finite menu and requires trans-model proposals that mix reliably.

Continuous discrepancy marginalization

Introduce discrepancy coefficients $c_a$ in a basis $e_a(f)$,

$$h(f;\theta,\mathbf c) =h_A(f;\theta)+\sum_a c_a e_a(f),$$

and marginalize

$$p(\theta\mid d) \propto \int d\mathbf c\, \mathcal L(d\mid\theta,\mathbf c) \pi(\theta)\pi(\mathbf c).$$

This can represent interpolation or calibration of a model error, but the basis and covariance must be trained or bounded using information independent of the analyzed noise. An excessively flexible discrepancy absorbs real signal structure; an excessively narrow one gives false confidence.

Injection-calibrated bias bounds

Analyze accurate simulations or held-out waveforms with the production approximant and measure repeated offsets or coverage. This directly tests the complete inference chain in the sampled region. Its weakness is support: a finite injection bank cannot guarantee accuracy outside the simulated masses, spins, eccentricities, modes, or detector conditions.

Results and interpretations at the stated cutoff

The following assessment is bounded at 2026-08-18. The official current method baseline is GWTC-5.0 Methods, §§ 2 and 4–5; it documents the waveform inputs and validation choices actually used for the release. It does not establish that any finite waveform set spans every physically relevant discrepancy.

Position	Strongest support	Main limitation	What would change the assessment
One validated approximant is adequate in a bounded domain	Agreement with numerical-relativity or perturbative targets and successful injections	Validation may miss weak modes, eccentricity, extreme spins, or detector-dependent weighting	Held-out injections showing bias or residual structure above the stated tolerance
Discrete model comparison captures practical uncertainty	Posterior stability across independently constructed families	Families share calibration data, analytic ingredients, and omissions	A genuinely independent model or discrepancy analysis broadening or shifting results
Flexible discrepancy models are more conservative	They can absorb smooth phase or amplitude errors absent from a finite model list	Prior and basis can erase identifiable physics or create arbitrary uncertainty	Predictive calibration on held-out waveforms across the claimed parameter support
Current differences are negligible for a particular event	Cross-model posteriors agree relative to statistical width	Agreement can be prior-driven or confined to reported summaries	Reanalysis with changed priors, modes, noise treatment, or a more informative detector network

The conservative conclusion is conditional: uncertainty propagation is an analysis design tied to a waveform domain and detector weighting, not a universal error bar. Model averaging cannot assign probability to a model that was never included.

Independent falsification checks

A useful campaign draws injections from a waveform source not used as the recovery model. It spans the claimed mass ratio, spin, inclination, mode content, and signal-to-noise range, repeats noise realizations, and records:

• bias normalized by posterior scale;
• credible-set coverage across injections;
• coherent residual signal-to-noise ratio;
• sensitivity to prior, frequency band, and PSD;
• evidence or predictive-score stability; and
• results with calibration and noise nuisance parameters enabled.

The test must compare at least two recovery families and a no-error control. If the same offset appears in zero noise and many independent noise realizations, it is evidence for repeatable misspecification rather than an unlucky fluctuation. A failure outside the injection support cannot be repaired by quoting an average mismatch within support.

Independent samplers, convergence measures, and injection campaigns test distinct parts of the event-inference chain Veitch et al. 2015, §§ III–V.

Common pitfalls

Treating two models as a probability distribution. Their spread is informative but has no automatic coverage interpretation. State why those models bracket the missing physics, or call the comparison a sensitivity test.

Adding errors in quadrature without a generative model. Statistical, calibration, and waveform errors can be correlated. Marginalize their joint nuisance model or demonstrate that the independence approximation is adequate.

Using the analyzed event to tune discrepancy freedom. This can absorb noise and then reuse the same data for inference. Training, prior calibration, and validation need a separation that makes the predictive claim testable.

Exercise

Consider a one-parameter waveform with tangent $u=\partial_\theta h$, Fisher information $\Gamma=(u\mid u)=25$, and discrepancy satisfying $(\delta h\mid u)=5$.

1. Find the leading systematic shift.
2. Compare it with the Fisher standard deviation.
3. What happens if the discrepancy is orthogonal to $u$ but has the same norm?

Solution

The bias formula gives

$$\Delta\theta_{\mathrm{sys}} =\Gamma^{-1}(\delta h\mid u) =\frac{5}{25}=0.2.$$

The local statistical standard deviation is $\sigma_\theta=\Gamma^{-1/2}=0.2$, so the systematic shift is one nominal standard deviation. If $(\delta h\mid u)=0$, the leading parameter bias vanishes even if $(\delta h\mid\delta h)$ is unchanged. The orthogonal discrepancy may instead remain as coherent residual power, demonstrating why mismatch magnitude and parameter bias are not interchangeable.

References

• Cutler, C., and Vallisneri, M. (2007). “LISA Detections of Massive Black Hole Inspirals: Parameter Extraction Errors Due to Inaccurate Template Waveforms.” Physical Review D 76, 104018. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). “GWTC-5.0: Methods for Identifying and Characterizing Gravitational-Wave Transients.” arXiv:2605.27224. Open PDF.
• Thrane, E., and Talbot, C. (2019). “An Introduction to Bayesian Inference in Gravitational-Wave Astronomy: Parameter Estimation, Model Selection, and Hierarchical Models.” Publications of the Astronomical Society of Australia 36, e010. DOI.
• Veitch, J., et al. (2015). “Parameter Estimation for Compact Binaries with Ground-Based Gravitational-Wave Observations Using the LALInference Software Library.” Physical Review D 91, 042003. DOI.