Using Gaussian Mixtures for $ b_j(O)$.

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

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


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

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

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

Baggenstoss 2017-05-19