Bayesian Evidence and Model Comparison

Bayesian evidence is the probability density of the observed data after averaging a model’s likelihood over its normalized prior measure. Evidence rewards fit but also accounts for how much prior parameter space predicted the data poorly. This page derives that tradeoff in a solvable Gaussian problem, distinguishes Bayes factors from likelihood ratios and posterior model probabilities, and states the prior, selection, numerical, and data records needed for a defensible gravitational-wave model comparison.

Required background. Probability and Statistical Inference supplies marginalization, Bayes’ theorem, normalized densities, and odds.

Helpful background. Gravitational Radiation and Waveforms supplies the competing waveform and polarization hypotheses used in applications.

Marginal likelihood and model odds

For model $\mathcal H$ with parameters $\theta$,

$$Z_\mathcal H \equiv p(d\mid\mathcal H) =\int_\Theta \mathcal L(d\mid\theta,\mathcal H) \pi(\theta\mid\mathcal H)\,d\theta.$$

The evidence $Z_\mathcal H$ normalizes the parameter posterior. For two models,

$$B_{10}=\frac{Z_1}{Z_0}$$

is the Bayes factor, and model posterior odds are

$$\frac{P(\mathcal H_1\mid d)} {P(\mathcal H_0\mid d)} =B_{10} \frac{P(\mathcal H_1)}{P(\mathcal H_0)}.$$

A Bayes factor is not a posterior model probability until model prior odds and the model set over which probabilities are normalized are specified. It is also not the maximum likelihood ratio: evidence averages over parameters rather than choosing their best-fitting values. The role of evidence and model priors in gravitational-wave inference is developed in Thrane and Talbot 2019, §3, Eqs. (10)–(21), PDF.

A Gaussian model-comparison calculation

Observe one scalar $x$ with known noise scale $\sigma$. Compare

$$\mathcal H_0:\mu=0, \qquad \mathcal H_1:\mu\sim\mathcal N(0,\tau^2),$$

with $x\mid\mu\sim\mathcal N(\mu,\sigma^2)$. Under the null,

$$Z_0 =\frac{1}{\sqrt{2\pi\sigma^2}} \exp\!\left(-\frac{x^2}{2\sigma^2}\right).$$

Under $\mathcal H_1$, convolution of the Gaussian likelihood and prior gives

$$Z_1 =\frac{1}{\sqrt{2\pi(\sigma^2+\tau^2)}} \exp\!\left[-\frac{x^2}{2(\sigma^2+\tau^2)}\right].$$

Therefore

$$B_{10} =\sqrt{\frac{\sigma^2}{\sigma^2+\tau^2}} \exp\!\left[ \frac{x^2\tau^2} {2\sigma^2(\sigma^2+\tau^2)} \right].$$

The exponential rewards improvement in fit. The square-root prefactor penalizes spreading prior mass over values of $\mu$ not supported by the datum. At $x=0$, widening $\tau$ monotonically lowers $B_{10}$ even though the maximum likelihood under $\mathcal H_1$ remains at the null value. This is a concrete prior-volume effect, not an arbitrary correction added after fitting.

The result provides two independent checks. As $\tau\to0$, $\mathcal H_1$ collapses to $\mathcal H_0$ and $B_{10}\to1$. At fixed nonzero $\tau$, sufficiently large $|x|/\sigma$ makes the fit improvement dominate. Any numerical evidence calculation for this model should reproduce the closed form over both regimes.

What evidence depends on

Evidence changes when a model-specific prior range changes because the model itself has changed. It also depends on:

• the normalized likelihood, including covariance-dependent determinants;
• parameter coordinates and Jacobians;
• waveform domain, frequency band, calibration, and PSD treatment;
• nuisance parameters shared or not shared between models;
• data and event selection; and
• model prior odds when reporting posterior odds.

An improper prior has no fixed normalization, so its ordinary evidence is undefined. A very broad but proper prior can make a flexible model predict the observed data diffusely and lower its evidence. Choosing bounds after seeing the event uses the data twice unless the selection is included in the model.

For nested models, identities such as the Savage–Dickey density ratio apply only under specific compatibility conditions on the priors and parameterization. They are not shortcuts around checking nuisance-prior matching or boundary behavior.

Gravitational-wave comparisons

Examples include signal versus noise, alternative waveform families, tensor versus enlarged polarization models, eccentric versus quasicircular binaries, or hypotheses with and without a beyond-general-relativity deviation. The interpretation is restricted to the declared model pair. A large $B_{10}$ between two inadequate waveform families does not establish that the favored family is absolutely adequate, and a null-favoring result cannot exclude deviations outside the parameterized alternative.

Selection must be conditioned consistently. For a single event, suppose the detection label $D$ is determined by the same data $d$ used in the likelihood. Let $\alpha_j=P(D\mid\mathcal H_j)$ be each model’s prior-predictive detection probability. Among selected data,

$$p(d\mid\mathcal H_j,D)=\frac{Z_j(d)}{\alpha_j}, \qquad \frac{P(\mathcal H_1\mid D)}{P(\mathcal H_0\mid D)} =\frac{\alpha_1}{\alpha_0} \frac{P(\mathcal H_1)}{P(\mathcal H_0)}.$$

Their product gives the original posterior odds $B_{10}P(\mathcal H_1)/P(\mathcal H_0)$, even when the two detection probabilities differ. Dividing the evidence by efficiency while keeping unselected model prior odds would be inconsistent. The same cancellation for single-event parameter inference is explained in Thrane and Talbot 2019, Appendix F, item 1, corrected PDF.

Population and rate inference retain selection terms because they model which and how many events enter the sample. Selection based on additional data omitted from $d$ also requires that information in the generative model. These cases are developed in Prior Measures and Reparameterization and Source Populations and Hierarchical Inference. The sample definition must be fixed before inspecting favorable Bayes factors.

Numerical evaluation and checks

Evidence is challenging because posterior samplers concentrate where the integrand is large while $Z$ depends on its normalization over the entire prior. Nested sampling estimates prior-volume compression; thermodynamic integration connects prior and posterior through a temperature path; bridge and importance methods use overlapping reference distributions. A posterior Markov chain alone does not supply evidence without an additional normalization method.

Every reported value should include the algorithm, software and commit, parameterization and normalized priors, live-point or temperature settings, stopping rule, random seeds, likelihood implementation, precision, repeated-run scatter, and controls on missed modes. Compare at least two settings or independent runs. The analytic Gaussian calculation above and an exactly enumerable low-dimensional case are useful unit tests. The LALInference paper describes nested-sampling and MCMC evidence validation in Veitch et al. 2015, §§III–V, PDF.

Observational boundary and reproduction

For an event analysis, pin the GWOSC or controlled data release, event version, detectors and GPS interval, calibration and PSD products, data-quality selections and gates, sample rate, frequency band, waveform versions, likelihood code, prior files, sampler configuration, and all excluded models or events. Report $\ln Z$ uncertainty and covariance when Bayes factors reuse runs. Calibration, PSD, waveform, and prior-bound alternatives should be propagated rather than represented by one favored number.

At the 2026-08-18 evidence cutoff, the O4b characterization choices and model-comparison settings are documented in LVK Collaboration 2026, §§4–5, PDF. They are dated analysis products. No catalog Bayes factor is recomputed or quoted here.

Common pitfalls

Maximum likelihood is not evidence. The best fit ignores the volume of parameter space that fits badly. Two models with the same maximum likelihood can have different evidences.

Bayes-factor thresholds are not universal laws. Interpretation depends on model completeness, prior odds, prior measure, data selection, and numerical uncertainty.

A missed mode biases normalization. Posterior summaries may appear stable while an isolated mode carrying appreciable prior-weighted likelihood is absent. Evidence needs mode-search and repeat-run checks.

Check your understanding

Set $\sigma=1$ and $\tau=3$ in the Gaussian comparison. Compute $B_{10}$ for (a) $x=0$ and (b) $x=3$. If the model prior odds are $P(\mathcal H_1)/P(\mathcal H_0)=1/9$, find the posterior odds for case (b).

Solution

Here $\sigma^2+\tau^2=10$, so

$$B_{10} =\frac{1}{\sqrt{10}} \exp\!\left(\frac{9x^2}{20}\right).$$

For $x=0$,

$$B_{10}=\frac{1}{\sqrt{10}}=0.31623\ldots ,$$

which favors the point null because the broad alternative spreads prior mass away from zero. For $x=3$,

$$B_{10} =\frac{e^{4.05}}{\sqrt{10}} =18.1507\ldots .$$

The posterior odds are approximately $18.1507/9=2.0167$ in favor of $\mathcal H_1$. Despite the Bayes factor near 18, the posterior model probability is only $2.0167/(1+2.0167)\simeq0.669$ because the prior odds favored $\mathcal H_0$.

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.