  We study the multiplicative version of the classical Furstenberg's filtering
problem, where instead of the sum $\mathbf{X}+\mathbf{Y}$ one considers the
product $\mathbf{X}\cdot \mathbf{Y}$ ($\mathbf{X}$ and $\mathbf{Y}$ are
bilateral, real, finitely-valued, stationary independent processes,
$\mathbf{Y}$ is taking values in $\{0,1\}$). We provide formulas for
$\mathbf{H}(\mathbf{X}\cdot\mathbf{Y}|\mathbf{Y})$. As a consequence, we show
that if $\mathbf{H}(\mathbf{X})>\mathbf{H}(\mathbf{Y})=0$ and $\mathbf{X}\amalg
\mathbf{Y}$, then $\mathbf{H}(\mathbf{X}\cdot
\mathbf{Y})<\mathbf{H}(\mathbf{X})$ (and thus $\mathbf{X}$ cannot be filtered
out from $\mathbf{X}\cdot\mathbf{Y}$) whenever $\mathbf{X}$ is not bilaterally
deterministic, $\mathbf{Y}$ is ergodic and $\mathbf{Y}$ first return to $1$ can
take arbitrarily long with positive probability. On the other hand, if almost
surely $\mathbf{Y}$ visits $1$ along an infinite arithmetic progression of a
fixed difference (with possibly some more visits in between) then we can find
$\mathbf{X}$ that is not bilaterally deterministic and such that
$\mathbf{H}(\mathbf{X}\cdot\mathbf{Y})=\mathbf{H}(\mathbf{X})$. As a
consequence, a $\mathscr{B}$-free system $(X_\eta,S)$ is proximal if and only
if there is always an entropy drop $h(\kappa\ast\nu_\eta)<h(\kappa)$ for any
$\kappa$ corresponding to a non-bilaterally deterministic process of positive
entropy. These results partly settle some open problems on invariant measures
for $\mathscr{B}$-free systems.
