Assignment task: Consider a variant of the LDS model, with a new latent transition that depends on an observed sequence of inputs
y1:T in the form:
zt+1 = Azt + Byt + wt
where matrix B is an additional model parameter and yt
is the observed input vector at time t. And
observation model
xt = Czt + Ddt + wt
where dt is also observed and D is an additional model parameter. How do the Kalman filtering and smoothing updates change for this variation? How about the EM-based parameter estimation procedure?