The prior of the states $\boldsymbol {S}$

Similarly, the prior of the states can be written as

$$\begin{split}p(\boldsymbol{S}\vert \boldsymbol{\theta}) &= p(... ... p(s_k(t+1)\vert\mathbf{s}(t), \boldsymbol{\theta}) \end{split}$$ (5.24)

where

$$p(s_k(t+1) \vert \mathbf{s}(t), \boldsymbol{\theta}) = N(s_k(t+1);\; g_k(\mathbf{s}(t)), \exp(2 v_{m_k}))$$ (5.25)

and

$$p(s_k(1) \vert \boldsymbol{\theta}) = N(s_k(1);\; m_{s^0_k}, \exp(2 v_{s^0_k})).$$ (5.26)