Let's say we have two random variables $X_1 \sim N(\mu,\sigma^2)$ and $X_2 \sim Bern(0.5)$. The binary outcome variable $Y$ is generated from \begin{align} P(Y=1|X_1,X_2)=\frac{e^{\beta_0+\beta_1X_1+\beta_2 X_2}}{1+e^{\beta_0+\beta_1X_1+\beta_2 X_2}} \end{align} $Y \sim Bern(P(Y=1|X_1,X_2))$
Is there simple way to derive the conditional distributions of $P(X_1|Y=1)$,$P(X_1|Y=0)$,$P(X_2|Y=1)$,$P(X_2|Y=0)$? I think both $X_1$ and $X_2$ will retain original distributions (Normal is still normal, Bernoulli is still Bernoulli).
We can make the conditional independence assumption if needed. i.e., $P(X_1,X_2|Y)=P(X_1|Y)P(X_2|Y)$.
Did you try the Bayes formula
$P(X_1 | Y_1) = P(Y \mid X_1) P(X_1)/P(Y)$
$P(X_1 | Y_1) = \frac{\sum_{X_2} P(Y \mid X_1, X_2) P(X_2) P(X_1)}{\int d X_1 P(Y \mid X_1 X_2) P(X_1) P(X_2)}$