  We consider the Cauchy problem for the defocusing cubic nonlinear
Schr\"odinger equation in four space dimensions and establish almost sure local
well-posedness and conditional almost sure scattering for random initial data
in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{3} < s < 1$. The main ingredient in the
proofs is the introduction of a functional framework for the study of the
associated forced cubic nonlinear Schr\"odinger equation, which is inspired by
certain function spaces used in the study of the Schr\"odinger maps problem,
and is based on Strichartz spaces as well as variants of local smoothing,
inhomogeneous local smoothing, and maximal function spaces. Additionally, we
prove an almost sure scattering result for randomized radially symmetric
initial data in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{2} < s < 1$.
