partially observed hidden markov model

Question

I am working on a data learning problem. Here's the framework: the data X_it for each observation i=1,...,N, time t=1,...,T, measures a bio-marker over time for an observation (continuous values). An binary variable Y_it (1 or 0), is a clinical attribute partially observed for each observation at each time point. There is quite a bit of missing data for X and Y. You can have missing data for both X_it and Y_it, sometime you have X_it but not Y_it. The goal: I want to be able to predict the values of Y_it of those observations where I know X_it. After some digging, i feel HMM is a good approach. But i stumble on a few difficulties: that is, there are some observed states Y_it that can be used to improve the model.However, i have no idea how to use them. Does anyone know a good method for this situation of mine? Thank you.