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This is easy. We
just fit each Gaussian to the data
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weighted by the
assignment probabilities that the
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Gaussian has for
the data.
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When
you fit a Gaussian to data you are maximizing
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the
log probability of the data given the Gaussian.
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This
is the same as minimizing the energies of the
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datapoints
that the Gaussian is responsible for.
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If
a Gaussian is assigned a probability of 0.7 for a
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datapoint
the fitting treats it as 0.7 of an observation.
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Since both the
E-step and the M-step decrease the
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same cost
function, EM converges.
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