Theory Priors and Parameter Measures

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A prior is part of a Bayesian model, not a decoration added after the likelihood. In gravity tests the choice is unusually consequential because phenomenological coefficients often vanish at the GR point, theory mappings are nonlinear or many-to-one, and stability or effective-field-theory conditions remove large parts of an apparently simple parameter space. A physical conclusion must therefore name both the parameter coordinates and the measure placed on them.

Required background. The Einstein Equation supplies the Einstein equations that define the reference point and physical solution space. Probability and Inference for Gravity supplies probability densities, transformations, marginalization, and posterior inference.

Helpful background. Weak Metric Perturbations and Gauge Freedom introduces common weak-field deformation coordinates. The Tests of Gravity overview shows how those coordinates recur across regimes.

A density is defined with respect to a measure

For parameters $\theta$ and data $d$ , Bayes’ rule reads

p(\theta\mid d,H) =\frac{p(d\mid\theta,H)\,\pi_\theta(\theta\mid H)} {\int p(d\mid\theta,H)\,\pi_\theta(\theta\mid H)\,d\theta}.

If $\phi=f(\theta)$ is a one-to-one reparameterization, probability conservation requires

\pi_\phi(\phi) =\pi_\theta\!\left(f^{-1}(\phi)\right) \left\lvert\frac{d f^{-1}}{d\phi}\right\rvert.

A density uniform in $\theta$ is generally not uniform in $\phi$ . Choosing a new uniform density after changing coordinates defines a new prior, not a harmless relabeling. For example, uniform support in a positive coupling $\alpha$ weights scales differently from uniform support in $\log\alpha$ . Neither is “uninformative” without reference to the experiment and the physical model.

This problem becomes visible near the GR point. Let $\alpha$ be uniform on $[-A,A]$ and $\delta=\alpha^2$ . The two inverse branches $\alpha=\pm\sqrt\delta$ contribute to the density, giving

\pi_\delta(\delta) =\frac{1}{2A\sqrt\delta},\qquad 0\lt\delta\lt A^2.

The apparent divergence at zero is integrable: $\int_0^{A^2}\pi_\delta\,d\delta=1$ . Conversely, a uniform density in $\delta$ gives a density proportional to $|\alpha|$ if the two signs receive equal weight. The observable and likelihood have not changed; the parameter measure has. A likelihood itself acquires no parameter Jacobian—one evaluates $\mathcal L_\alpha(\alpha)=\mathcal L_\delta(\alpha^2)$ —whereas transforming a prior or posterior density does require the branch Jacobians.

Support comes before weighting

Prior support should encode conditions required for the prediction to exist. These may include absence of ghosts and gradient instabilities, a well-posed initial-value problem, a selected background branch, positive masses, subluminal or otherwise admissible propagation under the adopted theory criterion, and an effective cutoff above every fitted scale. A likelihood value outside the controlled theory domain is not rescued by assigning it small probability.

Hard support and uncertain theory error should be separated. A mathematically inconsistent point receives zero support. A point with a controlled truncation error remains admissible, with the error represented in the forward model. A point whose nonlinear completion is unknown should not be reported as excluded by extrapolating a linear template through its breakdown.

“Naturalness” can motivate a scale but does not determine a unique probability distribution. If dimensional analysis suggests a coefficient $c_i/\Lambda^k$ , the prior requires an explicit range or hierarchy for $c_i$ and $\Lambda$ . Logarithmic ranges are especially sensitive to endpoints when the likelihood supplies only an upper bound. Reporting those endpoints is part of reporting the result.

Priors affect different summaries differently

When the likelihood is sharply concentrated far from boundaries, several smooth priors can yield similar parameter estimates. Gravity tests often inhabit the opposite limit: the deformation is consistent with zero, so the data provide an upper limit and the posterior retains information about the measure near the null.

Bayesian evidence is more sensitive still,

Z_H=\int \mathcal L(\theta)\,\pi(\theta)\,d\theta.

Widening the prior over regions of poor likelihood dilutes the average predictive density. That is the intended complexity penalty, but it means a Bayes factor cannot be quoted without the model support and measure. Evidence ratios and parameter-shift diagnostics also answer different consistency questions and should not be substituted for one another Raveri and Hu 2019, §II.C–E.

Frequentist procedures do not use parameter priors, but they still require a parameter space, boundaries, nuisance treatment, ordering rule, and coverage calibration. A confidence limit built with an asymptotic statistic can fail near a boundary or with an unidentified parameter; Monte Carlo calibration or a construction designed for the boundary is then required Particle Data Group 2024, “Statistics,” §§40.2–40.4.

A practical sensitivity analysis

A credible analysis should separate four choices:

Choice	Question to record	Useful check
Coordinates	Which quantities receive densities?	Transform one posterior with the Jacobian
Support	Where is the theory stable and predictive?	Recompute after enforcing each condition separately
Scale and shape	Why this linear, logarithmic, or hierarchical measure?	Vary justified scales and endpoints
Shared structure	Which experiments share hyperparameters?	Compare joint and falsely independent models

Prior sensitivity is not demonstrated by trying arbitrary distributions until a desired answer appears. Choose a defensible family before examining the decisive result, then report which scientific conclusions survive it. If an exclusion changes qualitatively across plausible measures, that dependence is the conclusion.

Hierarchical priors can express genuine population structure. Multiple gravitational-wave events may share a distribution of deviation parameters, or laboratory runs may share a calibration scale. The hierarchy must respect selection and should not force physically distinct regimes to share a universal coefficient unless a theory supplies that relation.

Check your understanding

An experiment gives a likelihood nearly constant for $0<\alpha<10^{-3}$ and rapidly decreasing above it. Compare qualitatively a prior uniform in $\alpha$ with one uniform in $\log\alpha$ on $[10^{-12},1]$ .

Solution

The linear prior assigns probability in proportion to interval length, so most of its surviving posterior mass lies near the largest values still allowed by the likelihood. The logarithmic prior assigns equal mass per decade, so the nine unconstrained decades below $10^{-3}$ dominate and the posterior depends strongly on the arbitrary lower endpoint. Therefore the two upper credible limits answer different prior-weighted questions even though the likelihood is identical. A responsible report shows the likelihood or a prior-robust sensitivity statement, names both endpoints, and avoids describing either posterior limit as an experiment-only fact.

For an exactly flat likelihood below $L=10^{-3}$ and zero above it, the 95% upper quantile is approximately $0.95L=9.5\times10^{-4}$ for the linear prior. For the logarithmic prior with lower endpoint $a=10^{-12}$ it is

\alpha_{95}=a\left(\frac{L}{a}\right)^{0.95} =10^{-3.45}\simeq3.55\times10^{-4}.

The tighter-looking second limit was produced by the measure assigned to unmeasured decades. It is not evidence that the second analysis had a more precise apparatus.

References

Particle Data Group (2024), “Statistics,” in Review of Particle Physics, Physical Review D 110, 030001. Particle Data Group 2024, §§40.1–40.4, PDF.
Raveri, M., and Hu, W. (2019), “Concordance and discordance in cosmology,” Physical Review D 99, 043506. Raveri and Hu 2019, §II.C–E.