Consider the logistic regression where $Y\_i \in {0,1}$ are dependent variable observations and $X\_i \in \mathbb{R}$ are the independent variables.

However we do not observe the $X\_i$ themselves. Instead we observe some vector of parameters $\boldsymbol{\mu}\_i$ and we know the distribution $F$ s.t. $F(X\_i|\boldsymbol{\mu}\_i)$.

How can we perform a logistic regression where the $X\_i$ are not themselves observed? For $F$ a general distribution, one way may be to sample for each $i$, many values from $F(X\_i|\boldsymbol{\mu}\_i)$ and put them all into the regression model as observations.

Can we do things more efficiently if we know for example that $F$ is a normal distribution?