Bayesian and Frequentist Inference

Gravitational-wave inference turns calibrated strain into conditional statements about source parameters, competing models, interval-producing procedures, and model adequacy. Those statements are not interchangeable. A likelihood describes data under parameters; a posterior adds a prior measure; evidence averages a likelihood over that measure; a confidence interval is calibrated across repeated data; and a predictive check tests whether the fitted model reproduces selected features. This chapter develops each object and the numerical checks that keep its interpretation honest.

Helpful background. Probability and Statistical Inference supplies conditional probability, change of variables, hypothesis tests, and Monte Carlo integration. Gravitational Radiation and Waveforms supplies the detector response and waveform parameters entering the likelihood.

The inferential map

For model $\mathcal H$, parameters $\theta$, and selected data $d$, begin with

$$\mathcal L(d\mid\theta,\mathcal H).$$

A normalized prior measure $\pi(\theta\mid\mathcal H)d\theta$ gives

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

The posterior answers a parameter question conditional on $\mathcal H$. The evidence $Z_\mathcal H$ compares prior-predictive performance between models through a Bayes factor. Neither defines a frequentist confidence procedure, whose coverage is a probability over repeated datasets at fixed truth. Finally, a posterior-predictive check simulates replicated data to ask whether chosen discrepancies are typical under the fitted model; it is not a posterior probability that the model is true.

These distinctions prevent common reversals:

• $p(d\mid\theta)$ is not $p(\theta\mid d)$;
• a small noise p-value is not a large signal posterior;
• a Bayes factor is not posterior model odds without model prior odds;
• a 90% credible region is not automatically a 90% confidence interval; and
• a non-extreme residual statistic does not validate every aspect of the model.

Route through the chapter

Read the pages in this order:

1. The Gravitational-Wave Likelihood derives the network Gaussian residual model and treats PSD, calibration, waveform, and data-selection assumptions.
2. Prior Measures and Reparameterization makes densities coordinate-aware through Jacobians and separates event priors from populations and selection.
3. Bayesian Evidence and Model Comparison derives marginal likelihood, prior-volume effects, Bayes factors, and model odds.
4. Posterior Sampling and Convergence compares Monte Carlo families and validates weighted, correlated, and multimodal exploration.
5. Frequentist Intervals and Coverage constructs repeated-experiment confidence sets and diagnoses boundaries and selection.
6. Posterior Predictive and Residual Checks tests model adequacy with whitened, coherent, null, and replicated discrepancies.

The first three define Bayesian objects. The fourth is numerical and serves any integral it is asked to approximate. The fifth develops a different calibration principle. The sixth challenges the adequacy of the model that all earlier conclusions condition upon.

The Gaussian control problem

One scalar measurement contains the chapter’s essential distinctions. Let

$$X\mid\mu\sim\mathcal N(\mu,\sigma^2), \qquad \mu\sim\mathcal N(0,\tau^2).$$

Then the likelihood is Gaussian in $X-\mu$, the posterior has

$$\operatorname{Var}(\mu\mid X) =\left(\sigma^{-2}+\tau^{-2}\right)^{-1},$$

the alternative-model evidence is $\mathcal N(X;0,\sigma^2+\tau^2)$, and the exact two-sided confidence interval is $X\pm z_{1-\alpha/2}\sigma$. A residual checked at a fixed true $\mu$ gives $(X-\mu)^2/\sigma^2\sim\chi^2_1$.

The formulas look similar because they share Gaussian geometry, but their conditioning differs. The prior scale $\tau$ changes posterior and evidence but not the exact known-$\sigma$ confidence rule. Fitting $\mu$ to $X$ changes a residual reference distribution. Selecting only large $X$ changes conditional coverage. This solvable case is the minimum control for a numerical implementation.

From the control to detector data

Real analyses replace one scalar by complex frequency bins across detectors and replace $\mu$ by masses, spins, distance, orientation, sky position, calibration, and waveform nuisance parameters. The Gaussian likelihood uses inverse-PSD weighting, but PSD is estimated and detector data contain lines, nonstationarity, and glitches. Priors live on nonlinear coordinates with bounds and symmetries. Posteriors can be multimodal, and evidence depends on tails and normalization. Detection and catalog selection can alter population likelihoods and selected-sample coverage.

The LALInference methods paper connects compact-binary likelihoods, priors, MCMC and nested sampling, validation, and coverage Veitch et al. 2015, §§II–V, PDF. Thrane and Talbot give a unified derivation of event and population inference, evidence, sampling, and selection Thrane and Talbot 2019, §§2–5 and appendices, PDF.

What the results establish and how to reproduce them

Every event result should pin the strain and product version, detectors and GPS interval, calibration and uncertainty representation, data-quality selection, gates, sample rate, window and frequency band, PSD estimator, waveform and likelihood commits, parameter coordinates and normalized priors, sampler configuration and seeds, convergence evidence, and posterior-sample weights. Model comparisons additionally need likelihood normalizations and model prior odds. Coverage and predictive checks need their full repeated-data or replicated-data generators.

The observational examples in this chapter have an evidence cutoff of 2026-08-18. The current O4b methods anchor is LVK Collaboration 2026, §§4–5, PDF. No page here recomputes a released event result. Analytic and synthetic controls isolate mathematical and numerical correctness; applying them to public data would additionally require pinned GWOSC files, hashes, calibration notes, and versioned analysis software.

Systematic uncertainty is not one catch-all number. Vary calibration response, PSD and conditioning, waveform family, prior measure and bounds, detector selection, and sampler configuration separately enough to identify what moves the result. Report failed fits and missed modes rather than filtering them from validation.

Review the chapter

1. Four statistical statements. For one event, an analysis reports a likelihood maximum at $\widehat\theta$, a 90% posterior credible interval, a Bayes factor of 12 for $\mathcal H_1$ over $\mathcal H_0$, and a residual-check tail of 0.3. State what extra information is needed to turn the Bayes factor into posterior model odds, and explain why the other three numbers answer different questions.

Solution

Posterior model odds require the model prior odds:

$$\frac{P(\mathcal H_1\mid d)}{P(\mathcal H_0\mid d)} =12\, \frac{P(\mathcal H_1)}{P(\mathcal H_0)}.$$

The likelihood maximum is a best-fitting parameter value under one data model and contains no parameter prior. The credible interval contains 90% posterior mass under its prior and likelihood. The Bayes factor compares prior-averaged predictive densities between two models. The residual tail says that the chosen discrepancy is not especially extreme in the chosen replicated ensemble; it is neither model odds nor a proof of adequacy.

2. Reparameterizing a posterior. A positive parameter has density $p_\theta(\theta\mid d)$ with $\phi=\ln\theta$. Derive the density in $\phi$. Does the posterior probability of $\theta\in[a,b]$ change?

Solution

Since $\theta=e^\phi$ and $d\theta/d\phi=e^\phi$,

$$p_\phi(\phi\mid d) =p_\theta(e^\phi\mid d)e^\phi.$$

The probability does not change:

$$\int_{\ln a}^{\ln b}p_\phi(\phi\mid d)\,d\phi =\int_a^b p_\theta(\theta\mid d)\,d\theta.$$

Density heights and the coordinate location of a mode may change, but probability assigned to the same physical region is invariant.

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.