  We continue the study on the transport properties of the Gaussian measures on
Sobolev spaces under the dynamics of the cubic fourth order nonlinear
Schr\"odinger equation. By considering the renormalized equation, we extend the
quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity.
Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia
[35] with the normal form approach in [30, 27].
