i've always just thought of kalman filters as continuous time/space equivalents of hmms, replacing multinomials with gaussians.
so yeah, you can do things like e-m to learn the parameters of the observation and state update models from data (just like fitting an hmm in the fully observed case), or given state and observation models, and past observations you can predict future state given current state. (what it was made for, where state is the state of your rocket given the physics of motion and all the noisy sensors or whatever, which if you think about it is like a viterbi search for the best fitting path through the continuous state space)
so yeah, you can do things like e-m to learn the parameters of the observation and state update models from data (just like fitting an hmm in the fully observed case), or given state and observation models, and past observations you can predict future state given current state. (what it was made for, where state is the state of your rocket given the physics of motion and all the noisy sensors or whatever, which if you think about it is like a viterbi search for the best fitting path through the continuous state space)