  We present a probabilistic cross-correlation approach to estimate time delays
in the context of reverberation mapping (RM) of Active Galactic Nuclei (AGN).
We reformulate the traditional interpolated cross-correlation method as a
statistically principled model that delivers a posterior distribution for the
delay. The method employs Gaussian processes as a model for observed AGN light
curves. We describe the mathematical formalism and demonstrate the new approach
using both simulated light curves and available RM observations. The proposed
method delivers a posterior distribution for the delay that accounts for
observational noise and the non-uniform sampling of the light curves. This
feature allow us to fully quantify its uncertainty and propagate it to
subsequent calculations of dependent physical quantities, e.g., black hole
masses. It delivers out-of-sample predictions, which enables us to subject it
to model selection and it can calculate the joint posterior delay for more than
two light curves. Because of the numerous advantages of our reformulation and
the simplicity of its application, we anticipate that our method will find
favour not only in the specialised community of RM, but in all fields where
cross-correlation analysis is performed. We provide the algorithms and examples
of their application as part of our Julia GPCC package.
