Bayes formula for conditional expectation
= Bayes formula for conditional expectation
If $d\widetilde{\mathbb P}=Z\,d\mathbb P$ and $Z_t=\mathbb E[Z\mid\mathcal F_t]>0$, then
$$
\widetilde{\mathbb E}[X\mid\mathcal F_t]
=\frac{\mathbb E[ZX\mid\mathcal F_t]}{Z_t}.
$$