Is it possible to fit a multivariate regression model where the independent variable is latent? I'm trying to fit a multivariate multiple regression model where the independent variable X is latent but I don't know where to start (I have prior information about the coefficient matrix so I can use some iterative method).
The dependent variable Y is a NxM matrix denoting N observations each from M variables. The latent variable X is a NxP matrix about which we know nothing except the dimensions. In addition to these, we have an initial estimate of the coefficient matrix beta based on prior knowledge. My goal is to find the estimates of latent X and coefficient matrix beta by both using the data matrix Y and the initial coefficient matrix. I thought of constructing an EM algorithm but because of the complexity of multivariate data and latent variable concept, I am totally confused.
Thank you.
 A: Might we reformulate the question as: 'I have N M-variate observations which I assume to be generated by N corresponding P-variate latent variables i.e. for each case/row M observed numbers are generated by P unobserved numbers.  I have an idea that this mapping is linear with an M x P matrix of coefficients and I want to know what the latent matrix values should be.'?
If that's accurate then you have a multivariate version of the regression calibration problem.  Normally one knows X and Y and estimates beta, whereas here one knows Y and beta and estimates / 'backs-out' X.  
This is what motivates suncoolsu's question about control - the question is about what distribution assumptions can be made about the marginal distribution of X (if any).  Your EM idea will make sense if you are happy to make distributional assumptions about P(Y | X; beta) and P(X) to apply Bayes theorem (although you won't need to iterate.) 
Or maybe that's not the problem you're facing and I just don't understand your description.
