An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models (1809.02293v9)

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.

• The paper introduces a comprehensive guide to applying Bayesian inference for parameter estimation and model selection in gravitational-wave astronomy.
• It explains computational techniques like MCMC and nested sampling to tackle high-dimensional data challenges such as autocorrelation and multimodality.
• The study highlights the use of hierarchical models to infer population properties, enhancing our understanding of astrophysical phenomena from gravitational-wave observations.

Overview of Bayesian Inference in Gravitational-Wave Astronomy

The paper by Eric Thrane and Colm Talbot provides a comprehensive introduction to Bayesian inference within the context of gravitational-wave astronomy. The focus is on parameter estimation, model selection, and hierarchical models, highlighting their role in the analysis of gravitational-wave data. The paper serves both as an educational resource for those new to Bayesian methods and a reference guide for more advanced practitioners.

Fundamentals of Bayesian Inference

The authors commence with a discussion on the basics of Bayesian inference, detailing how likelihoods, priors, and posteriors form the cornerstone of this statistical approach. The posterior distribution is central to Bayesian inference, allowing researchers to update their beliefs about model parameters in light of new data. For gravitational-wave studies, this involves characterizing the parameters of compact binary systems from observed data. Thrane and Talbot emphasize the importance of adequately choosing prior distributions, as these represent prior beliefs or empirical knowledge before observing the data.

Model Selection and Evidence

A critical component of Bayesian analysis is the evaluation of model evidence, enabling model selection. The authors explain the computation of Bayes factors—the ratios of evidences—to compare different models or hypotheses. The evidence, in this framework, is significant because it accounts for both the quality of fit and complexity of the models being evaluated. This balance is achieved through Occam's factor, penalizing models that are overly complex relative to the data.

Parameter Estimation via Sampling

The paper discusses practical computational techniques used in estimating Bayesian posteriors, including Markov Chain Monte Carlo (MCMC) and nested sampling. These methods address the challenges posed by the high dimensionality typical of gravitational-wave data. The authors provide an overview of these algorithms, highlighting issues such as autocorrelation and multimodality, as well as the advantages and limitations of each technique.

Hierarchical Models and Hyper-Parameters

With the increasing number of gravitational-wave detections, the need to paper population properties becomes essential. The paper elaborates on the use of hierarchical models and hyper-parameters to infer these population properties, such as the mass distribution of black holes. The derivation of the posterior predictive distribution (PPD) as an updated prior on model parameters is discussed, which allows integrating past data into new analyses.

Practical Implications and Future Directions

The paper is instrumental in demonstrating the practical applications of Bayesian inference for gravitational-wave astronomy. Through examples like determining the Hubble constant and analyzing black hole spins, it illustrates how these methods enable a deeper understanding of astrophysical phenomena. The Bayesian framework's flexibility in incorporating model uncertainties and selection effects is a critical advantage in extracting reliable physical insights from noisy and complex data.

In terms of future developments, the paper alludes to the potential advancements in Bayesian methods as gravitational-wave observations become more prevalent. This points towards refined models and more sophisticated computational techniques to handle large datasets and incorporate various sources of uncertainty more effectively.

Conclusion

Overall, Thrane and Talbot provide an essential resource for grasping the application of Bayesian inference in gravitational-wave astronomy. Their systematic presentation addresses foundational concepts while also tackling advanced topics like hierarchical modeling, making the paper a valuable reference for researchers seeking to expand their analytical toolkit in the field. Through detailed examples and methodologies, the work points to the critical role of Bayesian methods in advancing our understanding of the universe through gravitational-wave observations.