Gaussian mixture model question

hi, all
Assume the number of Gaussian models is known and I want to estimate
the parameters of the Gaussian models based on a given data set.
Assume each data only belongs to one class. Then based on the labeled
data, I can estimate the Gaussian parameters. If I get the optimized
labels, then the sum of determinants of the Gaussian covariant matrix
should be minimum, is that right? (It's intuitively right to me but I
don't know how to prove it) The expectation maximization algorithm is
looking for the optimization of log-likelihood. Then does optimization
of log-likelihood equals to the optimization of the sum of
determinants of the covariant matrixs? (By saying "equal" I mean: if I
take a snap shot of the labels at certain time during EM procedure
while the log-likelihood keeps increasing and then use that labeled
data to estimate the Gaussians, the sum of the determinants of the
Gaussian covariant matrces should be decreasing as the EM procedure
going on) Thanks for your comments. I'm not sure if I express it
clearly.
zl2k

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