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An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models

Eric Thrane, Colm Talbot

arXiv:1809.02293v9astro-ph.IM

TL;DR

The paper introduces Bayesian inference for gravitational-wave astronomy, covering parameter estimation, model selection, sampling, and hierarchical models. It develops the framework from likelihoods, priors, and posteriors through evidence and model comparisons, then extends it to population inference while discussing selection effects and sampler limitations.

  • Problem

    Bayesian inference provides the framework for extracting source and population information from gravitational-wave data, but its applications span parameter estimation, model selection, and hierarchical inference.

  • Method

    The paper presents an accessible primer covering likelihoods, priors, posteriors, evidence, model selection, sampling methods, and hierarchical models with gravitational-wave examples.

  • Results

    The paper organizes Bayesian inference into a unified treatment connecting individual-event posteriors, model comparisons, computational samplers, and population-level hierarchical models.

  • Takeaways & Limitations

    Bayesian inference supports gravitational-wave measurements ranging from source properties and cosmology to binary-formation and stellar-evolution studies.

  • Takeaways & Limitations

    Selection effects and MCMC autocorrelation require explicit treatment because they can affect hierarchical results and posterior-width estimates.

Abstract

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This is an introduction to Bayesian inference with a focus on hierarchical models and hyper-parameters. We write primarily for an audience of Bayesian novices, but we hope to provide useful insights for seasoned veterans as well. Examples are drawn from gravitational-wave astronomy, though we endeavor for the presentation to be understandable to a broader audience. We begin with a review of the fundamentals: likelihoods, priors, and posteriors. Next, we discuss Bayesian evidence, Bayes factors, odds ratios, and model selection. From there, we describe how posteriors are estimated using samplers such as Markov Chain Monte Carlo algorithms and nested sampling. Finally, we generalize the formalism to discuss hyper-parameters and hierarchical models. We include extensive appendices discussing the creation of credible intervals, Gaussian noise, explicit marginalization, posterior predictive distributions, and selection effects.

1 PREFACE: WHY STUDY BAYESIAN INFERENCE?

Bayesian inference connects gravitational-wave data to statements about astrophysical sources and the Universe. This section motivates the subject through concrete applications and positions the paper as an accessible primer.

  • Bayesian inference and parameter estimation support scientific statements about gravitational-wave sources and the Universe.
  • Gravitational-wave analyses use Bayesian inference to reconstruct merger locations and estimate source distances and masses.Examples include sky maps, GW170104’s distance, and GW150914’s component masses.
  • The framework is used to measure the Hubble constant, study binary-formation mechanisms, and investigate stellar death.
  • The paper provides a primer on Bayesian inference using gravitational-wave examples while aiming to remain understandable to a broader audience.
  • The review focuses on audioband gravitational waves from compact binary coalescences and points to applications in other gravitational-wave sources.

2 FUNDAMENTALS: LIKELIHOODS, PRIORS, AND POSTERIORS

Bayesian parameter inference combines a likelihood, prior, and evidence into a posterior distribution. The section explains how this distribution is interpreted, marginalized, and shaped by modeling choices and external information.

  • The posterior p(θ|d) is the probability density for model parameters θ given data d.For a small interval, p(θ′|d)dθ′ gives the probability that θ lies within that interval.
  • Bayes’ theorem combines the likelihood L(d|θ), prior π(θ), and evidence Z to form the posterior.
  • The likelihood describes the measurement and implicitly specifies a noise model; gravitational-wave analyses typically assume Gaussian detector noise.For complex data, the stated form corresponds to a two-dimensional Gaussian, or Whittle likelihood.
  • Priors encode beliefs before measurement, with choices ranging from isotropic sky-location priors to uniform or log-uniform distributions under ignorance.
  • Marginalization integrates over nuisance parameters to obtain lower-dimensional posteriors and incorporates uncertainty from correlated parameters.When parameters are covariant, the marginalized posterior can be broader than a conditional posterior at fixed parameter values.

3 MODELS, EVIDENCE AND ODDS

Bayesian evidence normalizes posteriors and enables model selection through evidence ratios, Bayes factors, and odds. These comparisons incorporate fit quality, prior volume, and prior beliefs about competing hypotheses.

  • The evidence Z is the fully marginalized likelihood and serves as the posterior’s normalization constant.
  • Model selection asks which model is statistically preferred by the data and by how much.
  • A Bayes factor is the ratio of evidences for two models, including signal-versus-noise and comparisons between distinct signal theories.
  • A threshold of |log BF| = 8 is often used to designate strong evidence for one hypothesis over another.The sign identifies which model is preferred.
  • Odds equal the Bayes factor multiplied by prior odds, which are often set to unity when competing hypotheses are initially treated as equally likely.
  • Evidence balances model fit against prior parameter volume, producing an Occam factor that penalizes excessive complexity.A simpler model can have greater evidence than a better-fitting model with a much larger prior volume.

4 SAMPLERS

The section introduces stochastic samplers for estimating high-dimensional posteriors, focusing on MCMC and nested sampling. It explains their outputs, uses, and practical limitations.

  • 4 SAMPLERS: Phenomenological waveforms make forward predictions straightforward, but calculating fifteen-parameter posteriors is difficult because of the curse of dimensionality.Stochastic samplers address this computational challenge.
  • 4 SAMPLERS: MCMC and nested sampling generate posterior samples, while some samplers also estimate the Bayesian evidence.Posterior samples support expectation values and marginalized visualizations such as corner plots.
  • 4.1 MCMC: MCMC walkers perform a random walk whose transition probabilities produce draws from the posterior distribution.Burn-in is required before accumulating samples, and termination depends on acquiring enough samples for an accurate representation.
  • 4.1 MCMC: MCMC chains can be autocorrelated, so thinning by the autocorrelation length is necessary to avoid underestimating posterior widths.Walkers may also miss separate posterior modes divided by low-probability regions; multiple starting points can mitigate this issue.
  • 4.2 Nested sampling: Nested sampling targets the evidence by iteratively replacing the lowest-likelihood live point with a higher-likelihood prior draw.Removed points receive prior-volume weights, and an upper evidence bound supplies a termination criterion.

5 HYPER-PARAMETERS AND HIERARCHICAL MODELS

Hierarchical Bayesian inference extends event-level inference to population properties through hyper-parameters that shape the prior distribution. The section develops the multi-event likelihood, posterior, evidence, recycling method, and selection-effect boundary.

  • 5 HYPER-PARAMETERS AND HIERARCHICAL MODELS: Hierarchical Bayesian inference studies population properties shared by multiple gravitational-wave events, such as neutron-star equations of state and black-hole mass distributions.Hyper-parameters describe the population prior rather than an individual event.
  • 5 HYPER-PARAMETERS AND HIERARCHICAL MODELS: The population hyper-posterior is obtained by marginalizing event parameters θ, multiplying the resulting likelihood by a hyper-prior π(Λ), and normalizing by the hyper-evidence.The hyper-evidence remains distinct from the ordinary event-level evidence.
  • 5 HYPER-PARAMETERS AND HIERARCHICAL MODELS: For N independent events drawn from one hyper-parameterized population, the total likelihood combines the individual event likelihoods.Hyper-models can be compared with Bayes factors, as in ordinary model selection.
  • 5 HYPER-PARAMETERS AND HIERARCHICAL MODELS: The full hierarchical integral has dimension 15N+M for binary black-hole events, making direct sampling increasingly difficult as the event count grows.Breaking it into event-level integrals through recycling reduces the computational burden.
  • 5 HYPER-PARAMETERS AND HIERARCHICAL MODELS: Event-level posterior samples can be recycled to evaluate the hierarchical likelihood, with the default sampling prior canceling from the final expression.This permits analyzing events individually before combining them at the population level.
  • 5 HYPER-PARAMETERS AND HIERARCHICAL MODELS: Selection effects are initially ignored, although detectability varies with parameters; for binary black holes, visible volume scales approximately as V ∝M^2.1.The appendix extends the method to accommodate these effects.

A CREDIBLE INTERVALS

Credible intervals summarize regions containing specified posterior probability. The section contrasts highest-posterior-density and symmetric constructions, emphasizing their different behavior for asymmetric, multimodal, or boundary-peaked distributions.

  • A CREDIBLE INTERVALS: A credible interval is a region of parameter space containing a specified fraction of posterior probability.A two-sigma credible region contains 95% of posterior probability, but that condition does not uniquely determine the region.
  • A CREDIBLE INTERVALS: The highest posterior density interval places a threshold so the posterior region above it contains the target probability.HPDI intervals can be asymmetric or multimodal and have minimum width.
  • A CREDIBLE INTERVALS: Symmetric credible intervals are constructed from the cumulative distribution function.They are simpler to construct than HPDIs, particularly from posterior samples.
  • A CREDIBLE INTERVALS: Symmetric intervals can mislead for multimodal distributions or distributions peaking near prior boundaries.These cases motivate considering the shape of the posterior when choosing an interval construction.
  • A CREDIBLE INTERVALS: Simulation testing checks calibration by verifying that the true parameter lies within the 90% credible interval 90% of the time.The test draws true parameters from the prior, simulates data, and then performs inference.

B GAUSSIAN NOISE LIKELIHOOD

The Gaussian-noise likelihood models Fourier-domain detector data using a complex Gaussian for each frequency bin. Independent bins and detectors combine multiplicatively, while the log-likelihood exposes noise-weighted matched-filter structure.

  • B GAUSSIAN NOISE LIKELIHOOD: The Gaussian-noise appendix represents detector strain data in the Fourier domain, with each frequency bin characterized by a single-sided power spectral density P(f).P(f) is proportional to strain squared and has units of Hz^-1.
  • B GAUSSIAN NOISE LIKELIHOOD: The single-bin likelihood compares data d_j with waveform template μ(θ) using frequency resolution Δf and a complex two-dimensional Whittle Gaussian.The normalization reflects the complex nature of the Fourier-domain data.
  • B GAUSSIAN NOISE LIKELIHOOD: Independent frequency-bin likelihoods multiply across the M bins, and multi-detector data add a detector index to the product.Combining detectors is mathematically analogous to combining frequency bins.
  • B GAUSSIAN NOISE LIKELIHOOD: The log-likelihood replaces products with sums and can be written using a noise-weighted inner product.A consistent omission of the normalization term leaves evidence ratios unchanged because the factor cancels.
  • B GAUSSIAN NOISE LIKELIHOOD: The expanded log-likelihood contains a noise-evidence term and terms involving matched-filter signal-to-noise ratios.This form connects parameter estimation with matched filtering, the maximum-likelihood technique for gravitational-wave detection.
  • B GAUSSIAN NOISE LIKELIHOOD: Reduced-order methods enable rapid evaluation of the likelihood function.This computational improvement supports practical inference with waveform models.

C EXPLICITLY MARGINALIZED LIKELIHOODS

The paper develops explicit likelihood marginalizations over coalescence time, phase, and luminosity distance to reduce inference costs and improve convergence. It also describes numerical implementations and the assumptions and edge effects that constrain these techniques.

  • Likelihood construction: Waveform templates are generated from intrinsic parameters and then projected onto the detector response using extrinsic parameters.The projection step is much faster than calculating the metric perturbation.
  • Motivation: Explicit marginalization over extrinsic parameters can reduce inverse-problem dimensionality, speeding calculations and improving convergence, especially for weak signals.The appendix marginalizes over coalescence time, phase, and/or luminosity distance.
  • Time marginalization: Coalescence-time marginalization discretizes time offsets and evaluates the resulting sum with a fast Fourier transform.The discretized prior is πk = π(t)∆t, and edge effects require a prior that prevents waveform wrap-around.
  • Phase marginalization: Phase marginalization uses a complex inner product and a uniform phase prior, yielding a likelihood involving a modified Bessel function.Its validity depends on assuming the signal is dominated by ℓ = 2, |m| = 2 modes.
  • Distance marginalization: Distance marginalization can be evaluated numerically and accelerated with a pre-computed lookup table, but cosmological redshift couples luminosity distance to lab-frame masses.The appendix extends the calculation by marginalizing over redshift and source-frame mass.

C.4 Distance marginalization with cosmological effects

Cosmological distance marginalization must account for the covariance between luminosity distance and observed lab-frame masses caused by redshift. The paper constructs numerical redshift integrations and lookup tables, while also explaining posterior reconstruction after marginalization.

  • Cosmological effects: Cosmological redshift makes lab-frame mass priors covariant with luminosity distance, so distance marginalization must include source-frame masses and redshift.A source-frame mass ms is observed with a redshifted lab-frame mass.
  • Prior transformation: The distance-marginalized likelihood is rewritten in lab-frame quantities after converting the source-frame prior into a lab-frame prior.The implied lab-frame prior follows from the chosen redshift and source-frame mass priors.
  • Numerical implementation: A redshift grid enables numerical marginalization and lookup tables for rapid evaluation while accounting for cosmological redshift.The implementation also requires a lookup table for π(z|ml).
  • Combined marginalization: When combining marginalizations, the correct order is to marginalize over phase before distance and then incorporate time marginalization with a fast Fourier transform.The paper warns that simultaneous implementation requires care with operation ordering.
  • Posterior reconstruction: Marginalized likelihoods can support population inference on source distances or redshifts, but they do not directly provide posterior samples for marginalized parameters.Those samples can be reconstructed by calculating conditional posteriors and drawing values numerically.

D POSTERIOR PREDICTIVE DISTRIBUTIONS

A posterior predictive distribution updates the population prior using the hyper-posterior learned from observed data. It describes the expected true parameters of a future event while distinguishing population uncertainty from event-level posterior uncertainty.

  • Definition: The PPD is the updated prior on parameters θ given the data and hyper-posterior p(Λ|d).It asks what the distribution π(θ) looks like after learning about the hyper-parameters.
  • Interpretation: The PPD represents the probability that the next event has true parameter values θ given the learned hyper-posterior.The Λ subscript distinguishes it from the event posterior p(θ|d).
  • Uncertainty: A single PPD does not show the variability in π(θ) caused by uncertainty in Λ.Plotting realizations π(θ|Λk) from hyper-posterior samples conveys that additional uncertainty.

E SELECTION EFFECTS

Selection effects arise because detectability varies across gravitational-wave events, so population inference must model the probability that signals exceed the detection threshold. The paper describes threshold-based, analytic, and injection-based ways to calculate this probability.

  • Motivation: Selection effects occur because closer, higher-mass, and face-on/off binaries are generally easier to detect, with additional effects from black-hole spin.Higher mass improves detectability only until the signal begins leaving the observing band.
  • Detection threshold: Detection is commonly defined by a matched-filter signal-to-noise threshold, usually ρth = 8 for one detector or ρth = 12 for a network of at least two detectors.The threshold is applied to signal-to-noise maximized over the relevant phase parameter.
  • Detection probability: The detection probability pdet is the probability that a signal exceeds the threshold, and it can be calculated from the signal-to-noise distribution.For fixed parameters, ρmf is normally distributed with mean ρopt and unit variance.
  • Visible volume: Alternatively, pdet can be expressed as the visible volume divided by the total spacetime volume, with visible volume typically estimated using injected signals.This provides a numerical route to selection-effect normalization.
  • Population inference: Selection-effect normalization enters population inference through the hyper-parameters and requires marginalizing over the Poisson-distributed event rate.The result also depends on the prior chosen for the rate R.

F ERRATUM

The erratum removes a flawed single-event selection-effects subsection and clarifies the distinction between odds and odds ratios. It explains that likelihood and prior selection factors cancel for detected single events.

  • The original single-event selection-effects subsection was removed because it contained formula errors and a conceptual error.The erratum specifically identifies errors in Eq. 89 and Eq. 95 of the arXiv version, corresponding to Eq. E2 and Eq. E8 in the published version.
  • For detected single events, the likelihood gains p_det^-1 while the detection-conditioned prior gains a compensating p_det factor.These factors cancel in the posterior, recovering the original no-detection likelihood.
  • The manuscript corrects eight references to “the odds ratio” because multiplying a Bayes factor by prior odds yields posterior odds.An odds ratio is a ratio of ratios, whereas odds are a ratio of probabilities.
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