Estimation of $g_i({\bf X})$

We assume that $p({\bf X}\vert\Omega_i)$ is multivariate Gaussian

$$p({\bf X}\vert\Omega_i) = \frac {1} {(2\pi)^{3/2}\vert\Sigma... ...} - {\bf {\mu}}_i)^t \Sigma_i^{-1} ({\bf X} - {\bf {\mu}}_i)]}$$ (28)

where ${\bf {\mu}}_i = (\mu_{i_1},\mu_{i_2},\mu_{i_3})^t$ is the 3-component mean vector and $\Sigma_i$ is the $3 \times 3$ covariance matrix. Therefore, $g_i({\bf X})$ is given as

$$\begin {array}{cc} {g_i({\bf X}) \: = \: - \frac {1} {2} \lo... ...:}\\ \log [P(\Omega_i)] \: - \: \log[p({\bf X})] \end {array}$$ (29)