Prior Measures and Reparameterization

A prior is a probability measure on physical possibilities, not a list of coordinate ranges. Its density changes when coordinates change, and its support encodes assumptions just as consequentially as its shape. This page develops the Jacobian rule, applies it to distance, orientation, and binary masses, separates event-level priors from population and selection models, and gives checks that prevent a sampler’s convenient coordinates from silently redefining the science question.

Required background. Probability and Statistical Inference supplies probability densities, change of variables, conditional probability, and Bayes’ theorem.

Helpful background. Gravitational Radiation and Waveforms supplies binary parameters, detector orientation, and redshifted waveform quantities.

A density belongs to a coordinate measure

Let $\Theta$ be a parameter space with prior probability measure $\Pi$. In coordinates $\theta$,

$$d\Pi=\pi_\theta(\theta)\,d\theta, \qquad \int_\Theta\pi_\theta(\theta)\,d\theta=1.$$

If $\phi=g(\theta)$ is one-to-one and differentiable, probability assigned to a physical region cannot depend on the label used for it. Therefore

$$\pi_\phi(\phi) =\pi_\theta\!\left(\theta(\phi)\right) \left\lvert\det\frac{\partial\theta}{\partial\phi}\right\rvert.$$

“Uniform” is consequently incomplete until the coordinate and bounds are named. A flat density in $\theta$ is generally not flat in $g(\theta)$. The role of a gravitational-wave prior is discussed in Thrane and Talbot 2019, §2, PDF; the Jacobian above follows directly by equating the probability of the same infinitesimal physical region in the two coordinates.

Posterior probabilities are coordinate invariant when the transformation is performed correctly:

$$p_\phi(\phi\mid d)\,d\phi =p_\theta(\theta\mid d)\,d\theta.$$

Posterior density heights and modes need not be invariant. Quoting only a one-dimensional mode without its coordinate can therefore be misleading.

Distance, redshift, and orientation

In a Euclidean ball of radius $D_{\max}$, a spatially uniform source distribution has

$$\pi(D)=\frac{3D^2}{D_{\max}^3}, \qquad 0\le D\le D_{\max}.$$

A prior uniform in $D$ assigns a different physical measure. Cosmologically, “uniform in comoving volume and source-frame time” gives a redshift measure proportional to

$$\pi(z)\,dz \propto \frac{1}{1+z}\frac{dV_c}{dz}\,dz,$$

within declared cosmology and redshift bounds. It is not exactly $D_L^2dD_L$ once the luminosity-distance–redshift relation and time dilation matter.

An isotropic direction is uniform in solid angle:

$$dP=\frac{d\Omega}{4\pi} =\frac{d\varphi}{2\pi}\frac{d(\cos\vartheta)}{2}.$$

Thus $\cos\iota$ is uniform on $[-1,1]$, while the inclination density is $\pi(\iota)=\tfrac12\sin\iota$ on $[0,\pi]$. A density uniform in $\iota$ overweights nearly face-on and face-off orientations relative to an isotropic population.

Binary-mass coordinates

Take ordered component masses $m_1\ge m_2\gt0$ and define total mass and mass ratio

$$M=m_1+m_2, \qquad q=\frac{m_2}{m_1}, \qquad 0<q\le1.$$

The inverse map is

$$m_1=\frac{M}{1+q}, \qquad m_2=\frac{Mq}{1+q}.$$

Its Jacobian is

$$\left\lvert \frac{\partial(m_1,m_2)}{\partial(M,q)} \right\rvert =\frac{M}{(1+q)^2}.$$

Therefore a density constant in $(m_1,m_2)$ becomes proportional to $M/(1+q)^2$ in $(M,q)$, subject to the transformed component-mass bounds. Drawing $M$ and $q$ independently from flat boxes does not reproduce a component-mass-uniform prior.

Waveform phase is often more directly informed by the detector-frame chirp mass

$$\mathcal M^\mathrm{det} =\frac{(m_1^\mathrm{det}m_2^\mathrm{det})^{3/5}} {(m_1^\mathrm{det}+m_2^\mathrm{det})^{1/5}}, \qquad m_i^\mathrm{det}=(1+z)m_i^\mathrm{source}.$$

A sampler may use $\mathcal M^\mathrm{det}$ for efficiency while the scientific prior is stated in source-frame component masses and redshift. The implementation must include the full Jacobian and bounds connecting those coordinates. The LALInference description gives concrete compact-binary parameter and prior choices in Veitch et al. 2015, §§II–III, PDF.

Bounds, labels, and singular coordinates

Prior support can dominate weakly measured parameters. Lower mass bounds, spin-magnitude caps, equation-of-state constraints, distance truncation, or inclination symmetries can place posterior mass against an edge. Every bound needs a physical or analysis rationale, and sensitivity to plausible alternatives should be reported.

Ordering $m_1\ge m_2$ removes label duplication but introduces a boundary at equal mass. Sampling two exchangeable masses without ordering covers each physical unequal-mass binary twice unless the normalization or label symmetry is handled. Angular coordinates have coordinate singularities at poles even when the physical measure is smooth. Densities should be checked by integrating probabilities over physical regions, not by inspecting a corner plot alone.

Improper priors may yield proper parameter posteriors in some estimation problems, but their arbitrary normalization makes ordinary marginal likelihoods and Bayes factors undefined. Model comparison requires normalized priors over every model-specific parameter.

Event priors, populations, and selection

An event-level analysis uses

$$p(\theta\mid d,\mathcal H) =\frac{\mathcal L(d\mid\theta,\mathcal H) \pi(\theta\mid\mathcal H)} {Z_\mathcal H}.$$

A population analysis instead assigns hyperparameters $\Lambda$ to a normalized source distribution $p(\theta\mid\Lambda)$ and accounts for the detection process through an efficiency integral. Posterior samples produced under an interim event prior can be reweighted, but the interim prior, population density, and selection function must remain distinct.

For a detection decision $D$ determined by the data already used in an event likelihood, $p(\theta\mid d,D)=p(\theta\mid d)$: once $d$ is known, the label $D$ supplies no extra information. This can also be seen by defining efficiency $\epsilon(\theta)=P(D\mid\theta)$ and $\alpha=\int\epsilon(\theta)\pi(\theta)d\theta$. For detected data,

$$p(d\mid\theta,D)=\frac{p(d\mid\theta)}{\epsilon(\theta)}, \qquad \pi(\theta\mid D)=\frac{\epsilon(\theta)\pi(\theta)}{\alpha}.$$

The two efficiency factors cancel in the posterior. Multiplying only the original event prior by efficiency, or dividing only its likelihood by efficiency, changes the problem incorrectly. This cancellation and a correction to the originally published single-event treatment are explained in Thrane and Talbot 2019, Appendix F, item 1, corrected PDF. In a population analysis, the parameter-dependent efficiency normalization remains essential; see Thrane and Talbot 2019, §5 and Appendix E, corrected PDF. Selection based on additional data omitted from the event likelihood must instead be modeled jointly.

Prior-predictive and implementation checks

Before conditioning on an event, draw parameters from the implemented prior and propagate them through the waveform and detector response. This prior pushforward reveals induced distributions that are obscure in sampling coordinates: detector-frame chirp mass, strain amplitude, signal duration, network response, or nominal signal-to-noise ratio. Adding noise from the declared likelihood produces a full prior-predictive data draw. An implausible concentration can expose a missing Jacobian, a frame conversion error, or interacting bounds even when every one-dimensional input histogram appears correct.

A prior-predictive distribution is not a forecast of the detected catalog unless event rates, observing time, detector noise, and the detection selection are also included. Unselected draws test the event prior and waveform implementation. Passing them through a search tests a specified population-and-selection model. Keeping those stages separate makes it possible to locate an error.

Implementation checks should integrate analytic marginals, compare empirical and expected cumulative probabilities, round-trip transformed coordinates, and evaluate normalization under every bound and ordering rule. For masses, test both unequal and equal-mass boundaries. For angles, check isotropy by uniform histograms in $\cos\iota$ and sky solid angle rather than in the polar angles themselves. For a sampler that operates in transformed coordinates, store both the scientific log prior and every log-Jacobian contribution; compare them at fixed test points across independent implementations.

Reproduction and current-analysis boundary

A reproducible prior record names the physical measure, coordinate density, normalization, support, ordering convention, source- versus detector-frame masses, cosmology, angular convention, spin parameterization, waveform restrictions, and every transformation implemented by the sampler. Unit tests should draw prior samples, transform them both directions, recover analytic marginals, check normalization numerically, and test boundary points.

For a real event, also pin the strain release and event version, detectors, calibration and PSD products, waveform and likelihood versions, sampler commit, and selection used to include the event. At the 2026-08-18 evidence cutoff, release-specific O4b priors and parameter coordinates are documented in LVK Collaboration 2026, §§4–5, PDF. This page uses that source only as a dated example; it does not prescribe those choices for all analyses.

Systematic checks should repeat inference under scientifically credible distance, mass, spin, cosmology, and waveform alternatives. Report which conclusions are likelihood dominated and which move with the prior. A sampler reproducing its own input prior is a necessary implementation check, not evidence that the prior is physically appropriate.

Check your understanding

Compare two normalized priors on $0\le D\le D_{\max}$: one uniform in distance and one uniform in Euclidean volume. What probability does each assign to $D\le D_{\max}/2$? If the likelihood is uninformative, what does this imply for the posterior?

Solution

For the distance-uniform prior,

$$P\!\left(D\le\frac{D_{\max}}2\right) =\int_0^{D_{\max}/2}\frac{dD}{D_{\max}} =\frac12.$$

For the volume-uniform prior,

$$P\!\left(D\le\frac{D_{\max}}2\right) =\int_0^{D_{\max}/2}\frac{3D^2}{D_{\max}^3}\,dD =\frac18.$$

If the likelihood is constant over the support, the normalized posterior equals the prior. The two analyses then assign probabilities $1/2$ and $1/8$ to the same physical region. This is not a coordinate paradox: the priors represent different physical measures.

References

• 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.