  We construct a class of variational methods for the study of open quantum
systems based on Gaussian ansatzes for the quantum trajectory formalism.
Gaussianity in the conjugate position and momentum quadratures is distinguished
from Gaussianity in density and phase. We apply these methods to a
driven-dissipative Kerr cavity where we study dephasing and the stationary
states throughout the bistability regime. Computational cost proves to be
similar to the truncated Wigner (TWA) method, with at most quadratic scaling in
system size. Meanwhile, strong correspondence with the numerically exact
trajectory description is maintained so that these methods contain more
information on the ensemble constitution than TWA and can be more robust.
