Probability and Inference for Gravity

Gravity data do not speak without a probabilistic model. A conclusion depends on what generated the data, which parameters and nuisance quantities were allowed, how uncertainty was represented, and how events entered the sample. This module checks that chain using small analytic models. It is an inference-literacy bridge, not a general statistics course and not a substitute for an experiment’s released likelihood.

Required background. Calculus and Linear Algebra for Relativity supplies normalization integrals, quadratic forms, and changes of variables.

You should leave able to construct and interpret likelihoods, priors, posterior comparisons, and uncertainties, and to diagnose selection and nuisance-model effects in a gravity example.

Probability and inference diagnostic

1. A datum $x$ obeys $x\mid\theta\sim\mathcal N(\theta,\sigma^2)$ and the prior is $\theta\sim\mathcal N(\mu_0,\tau^2)$. Derive the posterior mean and variance.
2. Explain why a $90\%$ Bayesian credible interval and a $90\%$ frequentist confidence interval do not make the same probability statement, even when their numerical endpoints happen to agree.
3. Let $y=\theta+c+n$, where $n\sim\mathcal N(0,\sigma^2)$ and an independent calibration offset $c\sim\mathcal N(0,s^2)$. Marginalize $c$ and identify the effective likelihood for $y\mid\theta$.
4. An intrinsic population has normalized density $p(z\mid\alpha)$ and detection probability $S(z)$. Write the density of $z$ among detected objects and identify the normalization that cannot be dropped when inferring $\alpha$.

Hints and solutions

Hints for 1

Multiply likelihood and prior, collect the quadratic terms in $\theta$, and complete the square. Precisions add.

Solution to 1

Bayes’ rule gives

$$p(\theta\mid x)\propto \exp\left[-\frac{(x-\theta)^2}{2\sigma^2} -\frac{(\theta-\mu_0)^2}{2\tau^2}\right].$$

Completing the square produces a Gaussian posterior with

$$V=\left(\frac1{\sigma^2}+\frac1{\tau^2}\right)^{-1}, \qquad \mu=V\left(\frac{x}{\sigma^2}+\frac{\mu_0}{\tau^2}\right).$$

As $\tau\to\infty$, the prior becomes broad and $(\mu,V)\to(x,\sigma^2)$. As $\sigma\to0$, the data dominate. These limits check the weights.

Hints for 2

Ask what is random in each statement. In Bayesian inference the posterior distribution describes uncertainty in the parameter conditional on the observed data; frequentist coverage concerns repeated data sets under fixed parameter values.

Solution to 2

A $90\%$ credible interval $C(x)$ satisfies

$$\Pr(\theta\in C(x)\mid x)=0.9$$

under the stated likelihood and prior. A $90\%$ confidence procedure is constructed so that, for each fixed true $\theta$, repeated data $X$ satisfy

$$\Pr_\theta[\theta\in C(X)]=0.9.$$

After observing one data set, the frequentist statement remains about the procedure’s repeated-sampling coverage, not a posterior probability assigned to fixed $\theta$. Coincident endpoints do not erase the difference in conditioning and interpretation.

Hints for 3

The sum of independent Gaussian variables is Gaussian. You can also complete the square in the integral $\int p(y\mid\theta,c)p(c)\,dc$.

Solution to 3

Since $c+n$ is the sum of independent zero-mean Gaussians,

$$y\mid\theta\sim\mathcal N(\theta,\sigma^2+s^2).$$

Thus

$$p(y\mid\theta)= \frac{1}{\sqrt{2\pi(\sigma^2+s^2)}} \exp\left[-\frac{(y-\theta)^2}{2(\sigma^2+s^2)}\right].$$

Ignoring calibration would overstate precision. A correlated or non-Gaussian calibration model would not reduce to this simple variance addition.

Hints for 4

Condition on the event “detected.” The numerator is the joint density of $z$ and detection; the denominator is the total detection efficiency under the population model.

Solution to 4

The detected density is

$$p(z\mid\alpha,{\rm det}) =\frac{S(z)p(z\mid\alpha)}{\xi(\alpha)}, \qquad \xi(\alpha)=\int S(z)p(z\mid\alpha)\,dz.$$

$\xi(\alpha)$ depends on the population parameters and therefore affects inference. Dropping it generally biases the inferred population toward regions that are easy to detect. If counts also carry information, the event-rate model and Poisson factor must be stated separately.

Interpret the result

• Secure: likelihood, prior, posterior, nuisance marginalization, and selection conditioning are kept distinct, with all distributions normalized in the relevant variables.
• Uncertain: the qualitative effect is right but one normalization or conditioning statement needs a hint. Repeat that problem with a changed variance or selection function.
• Not yet: a posterior is called a likelihood, a credible interval is given a coverage interpretation, nuisance uncertainty is fixed away without justification, or detected objects are treated as an unbiased population sample.

This diagnostic is low-stakes. A correct formula without its conditioning statement is not yet secure inference reasoning.

Focused review

Generative model and likelihood

Write the sequence that could generate the data: physical parameters, latent variables, instrument response, nuisance quantities, and noise. The likelihood is the data density conditional on those quantities; it is not a probability distribution over parameters until combined with a prior and normalized. The gravity-specific construction is developed in The Gravitational-Wave Likelihood.

Priors, posteriors, and checks

Bayes’ rule is $p(\theta\mid d)\propto p(d\mid\theta)\pi(\theta)$. A prior can encode a physical population, a coordinate measure, or regularization; those are different claims. Report sensitivity to consequential prior choices and check posterior predictive behavior, not only a parameter contour. See Compact-Binary Priors for a concrete domain where coordinate choices matter.

Nuisance parameters and selection

Marginalization propagates uncertainty; profiling optimizes over it and answers a different question. Selection conditions the observed sample and introduces an efficiency normalization that usually depends on population parameters. Selection Functions and Injection Campaigns develops these ideas for gravitational-wave detectors.

Comparison is conditional

A likelihood ratio compares named hypotheses at specified parameter values. A Bayes factor integrates each likelihood over its prior and is therefore prior-dependent. Neither number is a context-free probability that a physical theory is true. Calibration, waveform systematics, data selection, and model misspecification remain part of the inference.

Transfer task: calibration and thresholding

A measured amplitude is

$$\hat A=A+c+n,$$

with $n\sim\mathcal N(0,\sigma^2)$ and $c\sim\mathcal N(0,s^2)$ independent. An event is retained only if $\hat A\gt A_*$. Derive the marginal likelihood before selection, the detection probability at fixed $A$, and the conditional density for a retained measurement.

Hint 1

After marginalizing $c$, define $\Sigma^2=\sigma^2+s^2$. Standardize the threshold with $(A_*-A)/\Sigma$.

Hint 2

Conditioning truncates the Gaussian. Divide its ordinary density by the probability that a draw lies above the threshold.

Solution

Marginalization gives

$$p(\hat A\mid A)=\mathcal N(\hat A;A,\Sigma^2), \qquad \Sigma^2=\sigma^2+s^2.$$

If $\Phi$ is the standard-normal cumulative distribution,

$$P({\rm det}\mid A) =1-\Phi\left(\frac{A_*-A}{\Sigma}\right).$$

For an observed value above threshold,

$$p(\hat A\mid A,{\rm det}) =\frac{\mathcal N(\hat A;A,\Sigma^2)} {1-\Phi[(A_*-A)/\Sigma]}, \qquad \hat A\gt A_*.$$

It is zero below threshold. The denominator restores normalization and depends on $A$. Near threshold, retained noise fluctuations are preferentially positive, so treating the selected sample with the untruncated Gaussian biases amplitude inference upward. In a real gravity analysis, the selection variable, pipeline, population, and injection validation must all be specified.

Common errors

• Suppressing the normalization of a likelihood when it depends on the parameter being inferred.
• Treating a prior uniform in one coordinate as uniform after a nonlinear change of variables.
• Folding calibration uncertainty into error bars without representing its correlations.
• Correcting observed counts for selection while forgetting that the detected-parameter density is also conditioned.

Return to your route

After the transfer task, return to the route that sent you here: Compact Binaries and Gravitational Waves, Relativistic Astrophysics, Tests of Gravity, Relativistic Cosmology, Primordial Black Holes, Gravitational Radiation and Compact Binaries, Computational Relativity, or the Research Seminar in Gravitation.