Suppose I fit a linear model where Y ~ X1 + X2 + error. This model performs quite well. Now suppose I want to reverse this and estimate X1 and X2 given that I have observed Y.
I'm curious what is the most appropriate way to do this? Is it fitting two regressions where one of the variables (either X1 or X2) is latent? Is it some sort of parameter grid search where I have X1 ~ Y + X2 for a fixed X2 and I do a grid search over X2? (and then I reverse it and do the same for X1)?
Thank you for your suggestions!