Astrophysical and Analysis Priors

A prior is a probability measure, not a neutral list of allowed values. “Uniform” changes meaning under reparameterization, and compact-binary likelihoods often leave distance, orientation, spin, mass ratio, or tidal parameters weakly constrained. In those directions, visible posterior structure can be inherited from the analysis prior. The remedy is to state the measure, its support, and its physical purpose, then test whether conclusions survive scientifically reasonable alternatives.

Helpful background. Prior Measures and Reparameterization develops Jacobians and prior predictive checks; Binary Parameters and Observable Combinations identifies which directions are likely to remain weakly measured.

A prior includes its measure

If $x=g(y)$ is one-to-one, probability conservation requires

$$\pi_y(y) = \pi_x(g(y)) \left\lvert\frac{dg}{dy}\right\rvert.$$

Uniformity in $x$ therefore does not imply uniformity in $y$. For component masses, use $q=m_2/m_1\le1$ and write

$$m_1=\mathcal M(1+q)^{1/5}q^{-3/5}, \qquad m_2=qm_1.$$

The two-dimensional Jacobian is

$$\left\lvert \frac{\partial(m_1,m_2)}{\partial(\mathcal M,q)} \right\rvert =\mathcal M(1+q)^{2/5}q^{-6/5}.$$

Thus a density uniform in $(m_1,m_2)$ induces a density proportional to this expression in $(\mathcal M,q)$, inside the transformed component-mass bounds. The Jacobian alone does not replace those bounds: changing coordinates turns a rectangular or ordered mass domain into a curved one. Likewise, a prior uniform in spin magnitude differs from one uniform in Cartesian spin components.

The posterior under prior $\pi$ is

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

Changing $\pi$ changes both posterior and evidence. It is not legitimate to compare evidences computed under unstated or incompatible prior measures.

The standard compact-binary likelihood and explicit event-level prior construction are set out in Veitch et al. 2015, § II.

Common compact-binary choices

Distance and orientation

An isotropic orientation has

$$\pi(\cos\iota)=\frac12, \qquad -1\le\cos\iota\le1,$$

not a density uniform in $\iota$. A Euclidean constant-density approximation gives $\pi(D_L)\propto D_L^2$ inside finite bounds. For a population with constant merger rate per comoving volume and source-frame time, the corresponding cosmological prior is

$$\pi(D_L)\propto \frac{1}{1+z}\frac{dV_c}{dz} \left\lvert\frac{dz}{dD_L}\right\rvert.$$

It requires a distance–redshift relation and finite support. A prior uniform in comoving volume alone omits the source-rate time-dilation factor and is a different, permissible analysis choice; state which interpretation is intended.

Distance support can influence inclination because the likelihood has a broad distance–inclination ridge. Sky priors are uniform on solid angle, $d\Omega=d\alpha\,d(\sin\delta)$, unless external localization information is deliberately included.

Masses and spins

Analysis priors may be chosen for sampling efficiency or broad coverage—for example, bounded component masses, mass ratio, or detector-frame chirp mass. An astrophysical population prior answers a different question and belongs to a specified population model. In event-level inference, it can be useful to start with a broad interim prior and reweight later, provided posterior samples cover the target prior’s support.

Spin priors require choices for magnitude and direction. Isotropic directions are uniform in the azimuth and in the cosine of the tilt angle. A prior uniform in spin magnitude weights small magnitudes differently from a prior uniform in a three-dimensional spin ball, whose radial density scales as $\chi^2$.

Tides, eccentricity, and other extensions

Physical boundaries matter. Tidal deformabilities may be constrained to a model class; eccentricity depends on the reference frequency; and precession coordinates have singular or weakly identified limits. A broad numerical range is not automatically weakly informative in waveform space.

Diagnosing prior-driven structure

Several checks answer distinct questions:

1. Prior-to-posterior comparison. A marginal nearly identical to its prior is not strongly informed, although correlations may still be constrained.

2. Likelihood reweighting. If samples were drawn under interim prior $\pi_0$, expectations under $\pi_1$ can use weights

   $$w_k\propto\frac{\pi_1(\theta_k)}{\pi_0(\theta_k)},$$

   only where $\pi_0\gt0$ whenever $\pi_1\gt0$.

3. Effective sample size. A useful warning statistic is

   $$N_{\mathrm{eff}} = \frac{\left(\sum_k w_k\right)^2}{\sum_k w_k^2}.$$

   A tiny value means reweighting is dominated by a few samples and a new run may be required.

4. Prior predictive simulation. Draw $\theta\sim\pi$ and simulate waveforms or summary observables. This exposes a prior that concentrates on implausible detector-frame signals even when its one-dimensional marginals look broad.

5. Likelihood presentation. Where feasible, publish likelihood-reweightable samples or likelihood-informed summaries rather than describing one posterior as a direct measurement.

The event-level and hierarchical roles of interim priors are derived in Thrane and Talbot 2019, §§ 2–5 and appendices.

Analysis prior versus population statement

An event posterior computed with a convenient broad prior is not a draw from the astrophysical population posterior. Conversely, inserting a sharply structured population model into event inference can make a weak event appear individually informative. The clean separation is

$$\text{event likelihood} \longrightarrow \text{event posterior under stated interim prior} \longrightarrow \text{selection-normalized population analysis}.$$

Selection effects do not generally modify the likelihood for a single already-observed event under the usual conditioning, but they are essential when inferring rates or population distributions from a detected sample. The next chapter develops that normalization.

Dated observational boundary

As of 2026-08-18, prior choices for the current official release are documented in GWTC-5.0 Methods, §§ 4–5 and tied to the GWTC-5.0 data products. Those choices are release-specific analysis decisions, not timeless astrophysical distributions. Any numerical posterior claim must state the event version, waveform family, detector frame, cosmology, calibration and PSD treatment, prior support, and sampler convergence.

For a controlled injection, repeat the inference with at least two physically interpretable prior measures. Report both posterior displacement and $N_{\mathrm{eff}}$ for any reweighting. If the conclusion changes only because one prior excludes the injected value or leaves no sample support, the test diagnoses a support failure rather than evidence favoring a different source.

Exercises

1. Distance versus amplitude. Let amplitude $A=1/D$ with $D\in[D_{\min},D_{\max}]$, and take a Euclidean-volume prior $\pi_D(D)\propto D^2$. Find the induced density for $A$.

Solution

Since $D=1/A$ and $\lvert dD/dA\rvert=A^{-2}$,

$$\pi_A(A) \propto D^2\left\lvert\frac{dD}{dA}\right\rvert =A^{-2}A^{-2}=A^{-4},$$

on $A\in[1/D_{\max},1/D_{\min}]$. A prior uniform in distance, volume, or amplitude therefore represents three different assumptions.

2. Reweighting support. Why can samples from a prior restricted to $q\ge0.5$ not be reweighted to a target prior allowing $0.1\le q\lt0.5$?

Solution

The interim prior and hence the posterior sample distribution have zero density for $q\lt0.5$. Importance weights require the target density to be absolutely continuous with respect to the sampling density. No finite weight can create samples in missing support, so the target posterior requires a new inference run or a sampling distribution covering the full target domain.

References

• 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.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026). GWTC-5.0 Data Release. DOI 10.7935/bk00-6a89. Official release.
• 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. Open PDF.
• 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.