The Gravitational-Wave Likelihood

The gravitational-wave likelihood is a probability model for calibrated detector data conditional on a waveform, noise covariance, calibration response, and analysis selection. In its standard frequency-domain form it is a multivariate Gaussian in the residual between data and waveform. This page derives that form, identifies which normalization terms may be discarded and which may not, solves an amplitude example analytically, and states how PSD, calibration, waveform, and data-segment uncertainty limit the model.

Required background. Probability and Statistical Inference supplies conditional densities, multivariate Gaussians, covariance operators, likelihood ratios, and Bayes’ theorem.

Helpful background. Gravitational Radiation and Waveforms supplies detector strain models, polarizations, and source parameters.

Residuals and covariance

For detector $I$, write the selected calibrated strain as

$$d_I(t)=h_I(t;\theta)+n_I(t),$$

where $\theta$ contains source and nuisance parameters. The detector waveform is a delayed linear combination of the two tensor polarizations,

$$\widetilde h_I(f;\theta) =e^{+2\pi i f\,\Delta t_I} \left[ F_I^+(\alpha,\delta,\psi)\widetilde h_+(f;\theta) +F_I^\times(\alpha,\delta,\psi)\widetilde h_\times(f;\theta) \right].$$

Here $\Delta t_I$ is an actual arrival delay, so $h(t-\Delta t_I)$ acquires the displayed plus phase under $\widetilde x(f)=\int x(t)e^{+2\pi i f t}\,dt$. If instead a sky convention defines $\Delta t_I=-\boldsymbol n\cdot\boldsymbol x_I/c$, the equivalent phase is $e^{-2\pi i f\boldsymbol n\cdot\boldsymbol x_I/c}$.

For a finite vector of real data samples with zero-mean Gaussian noise covariance $C$, the normalized density is

$$p(d\mid\theta,C) =\frac{\exp\!\left[-\tfrac12 r^{\mathsf T}C^{-1}r\right]} {\sqrt{(2\pi)^N\det C}}, \qquad r=d-h(\theta).$$

Stationarity approximately diagonalizes $C$ in a Fourier basis. With a one-sided PSD $S_{n,I}(f)$ and positive frequencies, define

$$(a\mid b)_I =4\operatorname{Re}\int_{f_{\min}}^{f_{\max}} \frac{\widetilde a_I(f)\widetilde b_I(f)^*} {S_{n,I}(f)}\,df .$$

If detector noises are independent, the log likelihood becomes

$$\ln\mathcal L(d\mid\theta,S_n) =-\frac12\sum_I(d_I-h_I(\theta)\mid d_I-h_I(\theta))_I +\ln\mathcal N(S_n).$$

The normalizing term $\mathcal N$ is independent of $\theta$ only when the PSD, selected samples, and frequency bins are held fixed. It cancels in a likelihood ratio between waveforms evaluated on the same covariance, but not between different noise-covariance models. Comparing evidences computed from different data selections is more delicate: restoring the determinant alone does not make densities on different observation spaces comparable. Both hypotheses must describe the same observed data, with discarded samples marginalized or otherwise accounted for. The compact-binary likelihood and detector response are set out in Veitch et al. 2015, §§II.A–II.B, PDF and Thrane and Talbot 2019, Appendix B, PDF.

For a concrete normalization check, take a duration $T$ and positive-frequency bins separated by $\Delta f=1/T$, excluding DC and Nyquist. In the independent-bin approximation,

$$\mathbb E[\lvert\widetilde n_j\rvert^2] =\frac{T S_n(f_j)}2, \qquad p(\widetilde d_j\mid\theta,S_n) =\frac{2}{\pi T S_n(f_j)} \exp\!\left[-\frac{2\lvert\widetilde d_j-\widetilde h_j\rvert^2}{T S_n(f_j)}\right].$$

The density is with respect to $d\operatorname{Re}\widetilde d_j\,d\operatorname{Im}\widetilde d_j$: each quadrature has variance $T S_n/4$. Multiplying the bin densities produces the discrete version of $-\tfrac12(r\mid r)$ and makes the PSD-dependent normalization explicit.

What the Gaussian form assumes

The quadratic likelihood asserts more than “the histogram looks Gaussian.” It assumes:

• the selected residual is adequately described by a zero-mean distribution;
• its covariance is known or treated as a nuisance quantity;
• the Fourier bins used in the product are sufficiently independent;
• detector cross-covariances are negligible or modeled;
• the calibration map and waveform family are adequate over the chosen band; and
• the segment, gates, frequency limits, and quality cuts were selected without an unmodeled dependence on the inferred parameters.

Windowing correlates neighboring bins; PSD estimates fluctuate; spectral lines and glitches create non-Gaussian residuals; and detector state evolves in time. A Whittle likelihood is an approximation that becomes accurate for sufficiently long stationary records under regularity conditions, not an identity for every finite conditioned segment. The connection from calibration and detector noise to transient likelihoods is reviewed in LIGO Scientific Collaboration and Virgo Collaboration 2020, §§6–7 and 9.6, PDF.

An analytic amplitude posterior

Hold all waveform-shape parameters fixed and write $h=A u$. Define

$$D=(d\mid d),\qquad x=(d\mid u),\qquad H=(u\mid u).$$

Then

$$\ln\mathcal L(A) =-\frac12\left(D-2Ax+A^2H\right)+\text{constant}.$$

The maximum-likelihood amplitude and likelihood curvature are

$$\widehat A_\mathrm{ML}=\frac{x}{H}, \qquad -\frac{d^2\ln\mathcal L}{dA^2}=H.$$

With a normalized Gaussian prior $A\sim\mathcal N(0,\tau^2)$, completing the square gives a Gaussian posterior,

$$\sigma_A^2=\frac{1}{H+\tau^{-2}}, \qquad \mu_A=\sigma_A^2x.$$

This calculation checks the sign of the cross term and distinguishes three objects. The likelihood is a function of $A$ for fixed observed data. The maximum-likelihood estimator is a repeated-data statistic. The posterior is a probability density for $A$ conditional on the data and the declared prior measure.

Calibration and PSD uncertainty

Let the nominally reconstructed strain use response $R_{0,I}$ and define

$$C_I(f;\lambda_I) =\frac{R_{0,I}(f)}{R_{{\rm true},I}(f)} =\left[1+\delta A_I(f;\lambda_I)\right] e^{i\delta\phi_I(f;\lambda_I)}.$$

The waveform expressed in the nominal strain coordinates is

$$\widetilde h_I^\mathrm{obs}(f;\theta,\lambda_I) =C_I(f;\lambda_I)\widetilde h_I(f;\theta).$$

Indeed, if the detector error signal is $d_{\rm err}$, then $d_0=R_0d_{\rm err}=C_I[h(\theta)+n_{\rm true}]$. A PSD estimated from the released nominal stream already describes $C_In_{\rm true}$, so the model $C_Ih$ and that PSD are expressed in the same data coordinates. Calibration coefficients $\lambda_I$ may be sampled with priors supplied by the calibration analysis. They can correlate with luminosity distance, phase, sky position, and polarization. Replacing the response by its nominal curve is a conditional analysis, not marginalization over calibration uncertainty.

The PSD is also estimated from finite neighboring data. Plugging in one estimate treats it as exact and can understate uncertainty or shift evidence. Alternatives include jointly sampling PSD parameters, marginalizing scale uncertainty, or checking several off-source estimators. The same data must not be used in a way that silently tunes gates, PSD, and residual model to the candidate without accounting for that selection.

Observational boundary and reproduction

At the evidence cutoff of 2026-08-18, the GWTC-5.0 methods paper documents the O4b compact-binary likelihood, waveform inputs, priors, calibration treatment, sampler settings, and selected events LVK Collaboration 2026, §§4–5, PDF. These are versioned analysis choices; this page does not reproduce an event posterior or treat a named waveform release as timeless.

A reproducible event likelihood specifies the GWOSC or controlled data release, event version, detectors and GPS interval, sample rate, calibration product and uncertainty representation, data-quality mask, gates, window and FFT convention, frequency band, PSD data and estimator, waveform approximant and software commit, detector response convention, parameter coordinates, nuisance priors, likelihood implementation, and numerical precision. Retain strain and PSD hashes and record excluded detectors or frequencies with reasons.

Independent controls should recover the analytic amplitude posterior above, reproduce injected parameters in stationary Gaussian noise with calibrated coverage, compare residual spectra with the assumed PSD, vary PSD and calibration draws, and repeat with credible alternative waveform families. A synthetic signal delayed by known $\tau>0$ should acquire phase slope $+2\pi f\tau$ under the declared transform and recover the corresponding later arrival, providing a direct sign check. A good residual plot cannot establish that every waveform systematic is negligible; it tests the declared discrepancy summaries.

Common pitfalls

Dropping a normalization outside its domain. A covariance-dependent determinant can be ignored while varying only waveform parameters at fixed covariance. Noise-model comparisons require it; comparisons using different bins or data dimensions additionally require a common observation model.

Calling a likelihood a posterior. $\mathcal L(d\mid\theta)$ does not integrate to one over $\theta$ and contains no parameter measure. Multiplication by a normalized prior and division by evidence produces the posterior.

Treating the estimated PSD as known truth. The plug-in likelihood is conditional on that estimate. Finite PSD uncertainty and nonstationarity require marginalization or sensitivity checks.

Check your understanding

For a fixed waveform shape, let $x=(d\mid u)=12$, $H=(u\mid u)=4$, and use the prior $A\sim\mathcal N(0,1)$. Find the maximum-likelihood amplitude and the posterior mean and standard deviation.

Solution

The likelihood maximum is

$$\widehat A_\mathrm{ML}=\frac{x}{H}=3.$$

The posterior variance and mean are

$$\sigma_A^2=\frac{1}{4+1}=\frac15, \qquad \mu_A=\frac{x}{4+1}=\frac{12}{5}=2.4.$$

Thus $\sigma_A=1/\sqrt5\simeq0.4472$. The prior pulls the posterior mean toward zero relative to the likelihood maximum. With a broader prior, $\tau\to\infty$, the posterior approaches mean $3$ and variance $1/4$.

References

• LIGO Scientific Collaboration and Virgo Collaboration. “A Guide to LIGO–Virgo Detector Noise and Extraction of Transient Gravitational-Wave Signals.” Classical and Quantum Gravity 37 (2020): 055002. DOI. Open PDF.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Methods for Identifying and Characterizing Gravitational-Wave Transients.” arXiv:2605.27224 [gr-qc] (2026). Open PDF.
• Thrane, Eric, and Colm Talbot. “An Introduction to Bayesian Inference in Gravitational-Wave Astronomy: Parameter Estimation, Model Selection, and Hierarchical Models.” Publications of the Astronomical Society of Australia 36 (2019): e010. DOI. Open PDF, corrected arXiv v9.
• Veitch, John, et al. “Parameter Estimation for Compact Binaries with Ground-Based Gravitational-Wave Observations Using the LALInference Software Library.” Physical Review D 91 (2015): 042003. DOI. Open PDF.