[Paper Review] Fast Bayesian analysis of individual binaries in pulsar timing array data

B. Bécsy, Neil J. Cornish+1 more|arXiv (Cornell University)|Apr 14, 2022

Pulsars and Gravitational Waves Research48 references31 citations

TL;DR

This paper introduces a fast Bayesian method for analyzing individual supermassive black hole binaries in pulsar timing array data by separating signal parameters into shape and projection types. By precomputing inner products for shape parameters and efficiently marginalizing over projection parameters via multiple-try MCMC within a Metropolis-within-Gibbs scheme, the method accelerates Bayesian inference by over four orders of magnitude, enabling tractable analysis of complex models like eccentric binaries and multiple sources.

ABSTRACT

Searching for gravitational waves in pulsar timing array data is computationally intensive. The data is unevenly sampled, and the noise is heteroscedastic, necessitating the use of a time-domain likelihood function with attendant expensive matrix operations. The computational cost is exacerbated when searching for individual supermassive black hole binaries, which have a large parameter space due to the additional pulsar distance, phase offset and noise model parameters needed for each pulsar. We introduce a new formulation of the likelihood function which can be used to make the Bayesian analysis significantly faster. We divide the parameters into projection and shape parameters. We then accelerate the exploration of the projection parameters by more than four orders of magnitude by precomputing the expensive inner products for each set of shape parameters. The projection parameters include nuisance parameters such as the gravitational wave phase offset at each pulsar. In the new scheme, these troublesome nuisance parameters are efficiently marginalized over using multiple-try Markov chain Monte Carlo sampling as part of a Metropolis-within-Gibbs scheme. The acceleration provided by our method will become increasingly important as pulsar timing datasets rapidly grow. Our method also makes sophisticated analyses more tractable, such as searches for multiple binaries, or binaries with non-negligible eccentricities.

Motivation & Objective

• To address the computational bottleneck in Bayesian analysis of individual supermassive black hole binaries in pulsar timing array (PTA) data.
• To reduce the high computational cost arising from unevenly sampled, heteroscedastic PTA data and large parameter spaces.
• To enable efficient exploration of nuisance parameters such as pulsar distance, phase offsets, and noise models.
• To make advanced analyses—such as those involving multiple binaries or eccentric orbits—computationally feasible.
• To develop a method that scales efficiently with growing PTA datasets and complex signal models.

Proposed method

• The method separates signal parameters into 'shape' parameters (determining signal morphology) and 'projection' parameters (affecting line-of-sight projection).
• It precomputes expensive inner products for each shape parameter set, enabling rapid evaluation of the likelihood for any projection parameters.
• Projection parameters, including pulsar-specific phase offsets and noise amplitudes, are marginalized using multiple-try Markov chain Monte Carlo (MCMC) sampling.
• The approach uses a Metropolis-within-Gibbs scheme to iteratively sample shape and projection parameters, with projection parameters updated at negligible computational cost.
• The likelihood function is reformulated in the time domain to handle uneven sampling and non-stationary noise, avoiding Fourier-based approximations.
• The method is implemented in the QuickCW software package and supports full Bayesian inference without analytical maximization.

Experimental results

Research questions

• RQ1 How can Bayesian inference for individual supermassive black hole binaries in PTA data be accelerated without sacrificing accuracy?
• RQ2 Can the computational cost of exploring nuisance parameters like pulsar distance and phase offsets be reduced significantly?
• RQ3 To what extent can the new method scale with increasing dataset size and complexity of signal models?
• RQ4 Can the method enable efficient analysis of multiple binaries or eccentric orbits in PTA data?
• RQ5 How does the performance of the new method compare to traditional Bayesian approaches in terms of runtime and convergence?

Key findings

• The method accelerates Bayesian analysis of individual binaries by over four orders of magnitude through precomputation of inner products for shape parameters.
• Projection parameters, including pulsar phase offsets and noise amplitudes, are marginalized efficiently using multiple-try MCMC, enabling near-instantaneous updates.
• The approach maintains full Bayesian inference while avoiding the need for analytical maximization over nuisance parameters.
• The method scales sub-linearly with dataset size for projection parameter updates, making it suitable for large and growing PTA datasets.
• The method enables previously intractable analyses, such as searches for multiple binaries or eccentric systems, by reducing computational costs to manageable levels.
• Validation on simulated and real NANOGrav 11-year datasets confirms the method's accuracy and efficiency, with significant runtime reductions observed.