The Annals of Applied Statistics

Bayesian estimates of astronomical time delays between gravitationally lensed stochastic light curves

Hyungsuk Tak, Kaisey Mandel, David A. van Dyk, Vinay L. Kashyap, Xiao-Li Meng, and Aneta Siemiginowska

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The gravitational field of a galaxy can act as a lens and deflect the light emitted by a more distant object such as a quasar. Strong gravitational lensing causes multiple images of the same quasar to appear in the sky. Since the light in each gravitationally lensed image traverses a different path length from the quasar to the Earth, fluctuations in the source brightness are observed in the several images at different times. The time delay between these fluctuations can be used to constrain cosmological parameters and can be inferred from the time series of brightness data or light curves of each image. To estimate the time delay, we construct a model based on a state-space representation for irregularly observed time series generated by a latent continuous-time Ornstein–Uhlenbeck process. We account for microlensing, an additional source of independent long-term extrinsic variability, via a polynomial regression. Our Bayesian strategy adopts a Metropolis–Hastings within Gibbs sampler. We improve the sampler by using an ancillarity-sufficiency interweaving strategy and adaptive Markov chain Monte Carlo. We introduce a profile likelihood of the time delay as an approximation of its marginal posterior distribution. The Bayesian and profile likelihood approaches complement each other, producing almost identical results; the Bayesian method is more principled but the profile likelihood is simpler to implement. We demonstrate our estimation strategy using simulated data of doubly- and quadruply-lensed quasars, and observed data from quasars Q0957$+$561 and J1029$+$2623.

Article information

Ann. Appl. Stat. Volume 11, Number 3 (2017), 1309-1348.

Received: February 2016
Revised: January 2017
First available in Project Euclid: 5 October 2017

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Gravitational lensing microlensing Ornstein–Uhlenbeck process Gibbs sampler profile likelihood ancillarity-sufficiency interweaving strategy adaptive MCMC Q0957$+$561 J1029$+$2623 LSST quasar


Tak, Hyungsuk; Mandel, Kaisey; van Dyk, David A.; Kashyap, Vinay L.; Meng, Xiao-Li; Siemiginowska, Aneta. Bayesian estimates of astronomical time delays between gravitationally lensed stochastic light curves. Ann. Appl. Stat. 11 (2017), no. 3, 1309--1348. doi:10.1214/17-AOAS1027.

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Supplemental materials

  • R codes and data. This zip file [Tak et al. (2017)] contains all the computer code (Rcode.R) and data ( used in this article. An R package, timedelay, that implements the Bayesian and profile likelihood methods is publicly available at CRAN (