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

Abstract

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.

Measuring gravitational-wave memory in the first LIGO/Virgo gravitational-wave transient catalog

“…See Ref. [44] for a review of Bayesian statistics in the context of gravitational-wave astronomy. The total memory Bayes factor BF mem tot can then be accumulated over a series of N gravitational-wave observations,…”

The UTMOST pulsar timing programme – II. Timing noise across the pulsar population

“…In our analysis, the specific models we compared include: The specific choice of a Bayes factor threshold when performing model selection is largely dependent on what one considers to be an acceptable false positive rate. For instance, a conservative Bayes factor threshold of | ln(B12)| > 8 (corresponding to a false positive rate of ∼ 1/3000) is generally used in gravitational-wave astronomy (Thrane & Talbot 2019). A more common interpretation is outlined in Kass & Raftery (1995), where a Bayes factor of ln(B12) > 5 (false positive rate ∼ 1/150) is considered to be 'very strong' evidence for one hypothesis over the other.…”

Merger history of primordial black-hole binaries

“…. , d N ), which consist of N BBH merger events, we aim to extract the population parameters {θ, R 0 } from d. In order to do that, it is necessary to perform the hierarchical Bayesian inference on the BBHs' mass distribution [3,[61][62][63][64][65][66]. In this work, we will use the data of ten BBHs [3,7] reported by LIGO/Virgo O1 and O2 observations, and hence N = 10.…”

“…Because the standard priors on masses for each event in LIGO/Virgo analysis are taken to be uniform [3,7], the likelihood of an individual event p(d i |λ) is proportional to the posterior of that event p(λ|d i ). The total likelihood for an inhomogeneous Poisson process can be evaluated as [63][64][65][66] (15) where β(θ) is defined as…”