Autoregressive observations: HMM-AR

Conditional on the hidden variables $s=\{s_0,\dots,s_T\}$ the observations $x=\{x_1,\dots,x_T\}$ in the simple HMM, eq. (2), are independent. For time series modeling this independency is sometimes inadequate. An obvious extension is shown in Fig. 7. The $\psi$ function now becomes

$$\psi(s_t,x_t,s_{t-1},x_{t-1}) = p(x_t\vert x_{t-1},s_t)\,p(s_t\vert s_{t-1})$$ (31)

and the joint distribution is

$$p(xs) = \prod_{t=1}^T \psi(s_t,x_t,s_{t-1},x_{t-1})$$ (32)

where we define $\psi(s_1,x_1,s_0,x_0)=\tilde p(x_1\vert s_1)\,p(s_1\vert s_0)\,p(s_0)$ . Again, we have forward and backward recursions for the functions

$$\alpha_t(s) = p(x^t,s_t=s)$$ (33)

$$\beta_t(s) = p(x_{t+1}\dots x_T\vert s_t=s,x_t)$$ (34)

Fig.: HMM-AR conditional dependencies.