Use a single Gaussian to represent a mixture of Gaussians I want to merge a Gaussian mixture, $\sum_{i=1}^{K} w_i \exp(x; \mu_{i}, \Sigma_{i})$ into one single Gaussian, under the constraints that $w_1 &gt;&gt; w_2 \geq \dots \geq w_K$, i.e. we have a dominant Gaussian. Intuitively, the resulting Gaussian will be near the first Gaussian. I want to use a MLE estimate, seeking the optimal Gaussian $\exp(x; \mu, \Sigma)$. Obviously, formula for the expression of the mean is $$\mu = \sum_{i} w_i \mu_{i}$$
What about the covariance? Can any one help me with this problem?
One option is to re-estimate a single Gaussian distribution directly from the empirical data.