Simple Gaussian HMM

The hidden states are $s_t\in\{1,2,\dots,S\}$ and the observed variables are $x_t\in{\mathord{\mathbb{R}}}^d$ . The transition probabilities are that of a Markov chain, and the observation probabilities are Gaussian

$$p(s_t=i\vert s_{t-1}=j) = A_{ij}$$ (29)

$$p(x_t\vert s_t=i) = {p_{\mathcal{N}}}(x;\mu_i, V_i),\quad \mu_i\in{\mathord{\mathbb{R}}}^d, V_i\in{\mathord{\mathbb{R}}}^{d\times d}\quad .$$ (30)