Notation

The state variables are $s = (s_t)_{t=1,\dots,T}$ , the observed variables are $x=(x_t)_{t=1,\dots,T}$ . The observation probabilities are $p(x_t\vert s_t)$ and the transition probabilities of the unobserved (''hidden'') states are $p(s_t\vert s_{t-1})$ . Figure 1.1 shows the conditional dependency structure. The following abbrevations are used very frequently:

$$x^t = (x_1,x_2,\dots,x_t), \quad\mathrm{and}$$

$$x^{[t} = (x_t,x_{t+1},\dots,x_T)\quad .$$

Fig.: Basic conditional dependencies of Hidden Markov Models.

The $d$ -dimensional normal distribution is denoted as ${p_{\mathcal{N}}}(x;\mu,V), x\in{\mathord{\mathbb{R}}}^d$ ,

$${p_{\mathcal{N}}}(x;\mu,V) = (2\pi)^{d\over 2}\,\vert V\vert^{-{1\over 2}} \, \exp\left\{-{1\over 2}(x-\mu)^\prime\, V^- \,(x-\mu)\right\}$$ (1)