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Given a set of labels $y$ and design matrix $X$ we often compute a linear regression to find a set of parameters $\hat{\beta}$ such that $E[y|X] = X\hat{\beta}$. However, how does one perform regression when $X$ itself must be inferred conditioned on an observed set of data $O$?

Specifically, suppose we are given posterior probabilities for discrete values for each $x_i$ conditioned on $O$. Do there exist regression methods to take this uncertainty into account? A naive approach might just take the posterior mean for each $x_i$, but a more holistic approach may integrate over the posterior while performing the regression, that is find $\hat{\beta}$ for $E[E[y|X]|O]$.

If you know of references that would be great.

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The problem you have is called “measurement error” in your covariates. You observe the covariates with some error. This will lead to biased estimates (called the Attenuation bias) and fortunately there is a bunch of resources out there. I am listing a couple below:

http://web.stanford.edu/~doubleh/eco273B/survey-jan27chenhandenis-07.pdf

http://www.stat.tamu.edu/~carroll/talks/NCI_MEM_Call.pdf

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