Probability and Statistical Inference

Statistical inference turns data into statements only through a declared generative model. The likelihood, prior measure, selection rule, calibration and nuisance model, computational approximation, and diagnostic all contribute to the conclusion. This record distinguishes those objects and supplies the checks needed before a gravity result is interpreted; numerical constraints and model comparisons belong in dated empirical treatments.

Helpful background. Probability and Inference for Gravity supplies a bounded derivation of conditional probability, marginalization, interval interpretation, and selection effects.

Conditional objects and their questions

For data $d$, parameters $\theta$, nuisance quantities $\eta$, and model $\mathcal H$, begin from a normalized joint model. Its factors answer different questions:

Object	Definition or calibration	Interpretation
Likelihood	$\mathcal L(d\mid\theta,\eta,\mathcal H)$	Density of possible data at fixed parameters; not a probability density over $\theta$
Prior	$\pi(\theta,\eta\mid\mathcal H)$	Probability measure chosen before conditioning on these data
Posterior	$p(\theta,\eta\mid d,\mathcal H)\propto\mathcal L\pi$	Parameter uncertainty conditional on model, prior, and observed data
Evidence	$Z_{\mathcal H}=\int\mathcal L\pi\,d\theta\,d\eta$	Prior-predictive density used in model comparison
Confidence procedure	Repeated-data coverage at fixed truth	Calibration of a rule, not posterior mass for one realized interval
Predictive check	Replicated data and a declared discrepancy	A test of selected aspects of model adequacy, not the probability that the model is true

These distinctions and their gravitational-wave realization are developed in Thrane and Talbot 2019, §§ 2–3. Posterior model odds additionally require model prior odds; a Bayes factor alone is not a context-free probability for a theory.

Measures, transformations, and nuisance quantities

A density is coordinate-dependent even when probability is not. For a one-to-one change between one-dimensional parameters, $\phi=f(\theta)$,

$$p_\phi(\phi\mid d) =p_\theta(\theta(\phi)\mid d) \left\lvert\frac{d\theta}{d\phi}\right\rvert.$$

Therefore “uniform prior” is incomplete without a coordinate and support. Boundaries, symmetries, quotient identifications, and singular Jacobians must be explicit. Marginalization integrates nuisance uncertainty with a probability model; profiling maximizes over nuisance quantities and answers a different question. Calibration response, noise spectrum, waveform error, foregrounds, and population hyperparameters should remain represented until their correlations and impact have been tested.

Selection and hierarchical inference

Let $\Lambda$ describe an intrinsic population $\pi(\theta\mid\Lambda)$ and let $p_{\mathrm{det}}(\theta)$ be the probability that an object with parameters $\theta$ enters the sample. The detection efficiency is

$$\xi(\Lambda) =\int p_{\mathrm{det}}(\theta)\pi(\theta\mid\Lambda)\,d\theta.$$

Conditioning on $N$ detected events gives the schematic population likelihood

$$\mathcal L(\{d_i\}\mid\Lambda,N,{\rm det}) \propto \xi(\Lambda)^{-N} \prod_{i=1}^{N} \int \mathcal L_i(d_i\mid\theta) \pi(\theta\mid\Lambda)\,d\theta.$$

Dropping $\xi(\Lambda)$ generally biases the population toward easy-to-detect regions. If event counts carry information, a rate model and Poisson factor replace the conditional-on-$N$ convention and must be stated. Posterior-sample recycling also divides out each event’s interim prior before applying the population measure. These selection and hierarchy rules are derived in Thrane and Talbot 2019, §§ 4–5 and appendices.

Diagnostics before interpretation

Risk	Minimum discriminating check
Sampler misses mass or a mode	Effective sample size or weight diagnostics, multiple starts, repeated runs, and a solvable control
Prior or coordinate drives the result	Reweight or rerun consequential alternatives and compare probabilities of the same physical regions
Calibration or waveform uncertainty is fixed away	Marginalize a versioned nuisance model and vary its support or correlation structure
Selection model is wrong	Injection recovery over the population support, with pipeline and threshold versions pinned
Interval is misinterpreted	State credible mass or repeated-sampling coverage and test the claimed calibration
Fitted model is inadequate	Posterior-predictive or residual checks using discrepancies not exhausted by the fit

Compact-binary analyses provide a useful end-to-end example: strain segment, detector response, noise estimate, waveform, coordinates, normalized priors, sampler settings, convergence tests, injections, and coverage are one inferential chain Veitch et al. 2015, §§ II–V. A stable trace or one evidence estimate is not enough; report failed runs and sensitivity to analysis choices.

For detector-level likelihoods and interval procedures, continue to Bayesian and Frequentist Inference. Every reported bound, rate, event probability, or model comparison should identify the data version, selection, calibration, prior, uncertainty convention, software and waveform versions, diagnostics, and evidence cutoff.

References

• Thrane, Eric, and Colm Talbot. 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, John, 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. Open PDF.