Using Gaussian Mixtures for $b_j(O)$.

It will be convenient to model the PDF's $b_{j}(O_t)$ as Gaussian mixtures:

$$b_j(O) = \sum_{m=1}^M \; c_{jm} \; {\cal N}(O,$$$$\mbox{\boldmath$\mu$}$$$$_{jm},{\bf U}_{jm}), \;\;\;\; 1\leq j \leq N$$

where

$${\cal N}(O,$$$$\mbox{\boldmath$\mu$}$$$$_{jm},{\bf U}_{jm}) = (2\pi)^{-{P}/2} \vert{\bf U}_{jm}\vert^{-1/... ...mu$}_{jm})^\prime {\bf U}_{jm}^{-1} (O - \mbox{\boldmath$\mu$}_{jm}) \right\},$$

and $P$ is the dimension of $O$. We will refer to these Gaussian mixture parameters collectively as

$$b_j \stackrel{\mbox{\tiny$\Delta$}}{=}\{ c_{jm}, \mbox{\boldmath$\mu$}_{jm}, {\bf U}_{jm} \}.$$