Max-likelihood inference and EM-Algorithm

The full joint pdf of observed and hidden variables is

$$p(xs) = p(s_1)\prod_{t=2}^T p(s_t\vert s_{t-1}) \prod_{t=1}^T p(x_t\vert s_t) \quad .$$ (2)

The M-step in the EM algorithm amounts to finding

$$\max_p \int ds\, p^\mathsf{old}(s\vert x) \,\ln p(xs)$$ (3)

and via eq (2) this is just

$$\max_p \int ds\, p^\mathsf{old}(s\vert x) \left[\ln p(s_1)+\sum_{t=2}^T \ln p(s_t\vert s_{t-1}) + \sum_{t=1}^T \ln p(x_t\vert s_t) \right]\quad .$$ (4)

The factorization allows to maximise the three terms inpedendently,

$$\max_p \int ds_1\, p^\mathsf{old}(s_1\vert x) \ln p(s_1),$$

$$\max_p \sum_{t=2}^T \int ds_{t-1} ds_t\, p^\mathsf{old}(s_{t-1} s_t\vert x) \ln p(s_t\vert s_{t-1}),$$

$$\max_p \sum_{t=1}^T \int ds_t\, p^\mathsf{old}(s_t\vert x) \ln p(x_t\vert s_t) \quad ,$$

i.e. the quantities $p(s_{t-1} s_t\vert x)$ and $p(s_t\vert x)$ have to be calculated. Introduce the following shorter notation for these:

$$\gamma_t(s) = p(s_t=s\vert x)\quad,$$

$$\Omega_t(s,s^\prime) = p(s_{t-1}=s,s_t=s^\prime\vert x)\quad.$$ (5)