  We prove a weak rate of convergence of a fully discrete scheme for stochastic
Cahn--Hilliard equation with additive noise, where the spectral Galerkin method
is used in space and the backward Euler method is used in time. Compared with
the Allen--Cahn type stochastic partial differential equation, the error
analysis here is much more sophisticated due to the presence of the unbounded
operator in front of the nonlinear term. To address such issues, a novel and
direct approach has been exploited which does not rely on a Kolmogorov equation
but on the integration by parts formula from Malliavin calculus. To the best of
our knowledge, the rates of weak convergence are revealed in the stochastic
Cahn--Hilliard equation setting for the first time.
