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 ap-
pear in the sky. Since the light in each gravitationally lensed image
traverses a different path length from the quasar to the Earth, fluc-
tuations 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 ex-
trinsic 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 likeli-
hood of the time delay as an approximation of its marginal posterior
distribution. The Bayesian and profile likelihood approaches comple-
ment 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
.