Bayesian Estimates of Astronomical Time Delays between Gravitationally
  Lensed Stochastic Light Curves
  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.
