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