# regress on to a design matrix sampled from a distribution

Just wondering if I am thinking of this correctly. I would really appreciate any comments on the approach.

I have a problem where I need to regress the outcome ($$y$$) on a design matrix ($$X$$). So,

$$y | X \sim Normal(X\beta, \sigma)$$

However, I don't directly observe $$X$$ and rather I know that it comes from a distribution, so that:

$$X \sim Normal(\hat{\mu}, \hat{\tau})$$

Here, I have done some abuse of notation. $$\hat{\mu}$$ is a matrix and $$\hat{\tau}$$ is not really a covariance matrix, rather each column of $$\hat{\tau}$$ is a diagonal covariance matrix.

Perhaps a clearer way to writing would be:

$$X[:,i] \sim Normal(\hat{\mu}[:,i],\hat{\tau}[:,i])$$

borrowing from MATLAB syntax.

My question is that does the following model make sense:

$$X \sim Normal(\hat{\mu}, \hat{\tau})$$

$$\beta \sim Dirichlet(\textbf{1})$$

$$y|X \sim Normal(X\beta, \sigma)$$

where $$\hat{\mu}$$ and $$\hat{\tau}$$ are known and we try to estimate $$\beta$$ (and $$\sigma$$).

If yes, what sort of model is this? Is this a generative model?

The reason this formulation confuses me is that since $$X$$ is not observed, does it make sense to regress onto a distribution of the design matrix?

Thank you for any suggestions!