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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!

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It seems like you'd be interested in reading about Bayesian hierarchical modeling, as that seems well suited for your kind of problem where you know something about the hyperparameters and hyperpriors. I don't know too too much about it, but I'm sure there's loads of stuff online to better understand it, and your problem seems like a classic problem it deals with.

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