Full-Covariance Gaussian

Definition

A full-covariance Gaussian variational approximation is a multivariate Gaussian with an unrestricted covariance matrix:

$$q_\phi(z)=\mathcal N(z\mid \mu,\Sigma),$$

where $\Sigma$ is not constrained to be diagonal.

Why it matters in Q1

It can represent tilted elliptical contours and posterior dependence, unlike a diagonal Gaussian.