  The appearance of rogue waves in deep sea is investigated using the modified
nonlinear Schr\"odinger (MNLS) equation in one spatial-dimension with random
initial conditions that are assumed to be normally distributed, with a spectrum
approximating realistic conditions of a uni-directional sea state. It is shown
that one can use the incomplete information contained in this spectrum as prior
and supplement this information with the MNLS dynamics to reliably estimate the
probability distribution of the sea surface elevation far in the tail at later
times. Our results indicate that rogue waves occur when the system hits
unlikely pockets of wave configurations that trigger large disturbances of the
surface height. The rogue wave precursors in these pockets are wave patterns of
regular height but with a very specific shape that is identified explicitly,
thereby allowing for early detection. The method proposed here combines Monte
Carlo sampling with tools from large deviations theory that reduce the
calculation of the most likely rogue wave precursors to an optimization problem
that can be solved efficiently. This approach is transferable to other problems
in which the system's governing equations contain random initial conditions
and/or parameters.
