  This paper presents the convergence analysis of the spatial finite difference
method (FDM) for the stochastic Cahn--Hilliard equation with Lipschitz
nonlinearity and multiplicative noise. Based on fine estimates of the discrete
Green function, we prove that both the spatial semi-discrete numerical solution
and its Malliavin derivative have strong convergence order $1$. Further, by
showing the negative moment estimates of the exact solution, we obtain that the
density of the spatial semi-discrete numerical solution converges in
$L^1(\mathbb R)$ to the exact one. Finally, we apply an exponential Euler
method to discretize the spatial semi-discrete numerical solution in time and
show that the temporal strong convergence order is nearly $\frac38$, where a
difficulty we overcome is to derive the optimal H\"older continuity of the
spatial semi-discrete numerical solution.
